I. Interpretation of the Paper’s Core Abstract

Melt extrusion additive manufacturing (MEX) is the mainstream process for fabricating complex silicone rubber structures in medical and robotics applications; however, the lack of a quantifiable model for predicting extrusion force severely constrains process stability and part‑level accuracy. This study develops two practical predictive models that comprehensively distinguish between two primary sources of force—internal nozzle pressure thrust \(F_N\) and interlayer back pressure \(F_C)—and jointly compute the total extrusion force \(F_T\).

Using Dow Corning 737 silicone rubber as the printing material, a custom MEX‑based printing system equipped with piezoresistive sensors was employed to conduct multivariate experiments, validating the accuracy of two modeling approaches: the semi‑analytical model achieved prediction accuracies ranging from 89.7% to 99.7%, while the CFD numerical model delivered accuracies between 95.4% and 99.3%. Testing across three key process parameters—nozzle diameter, volumetric flow rate, and layer thickness—revealed that lower layer thicknesses result in higher back pressures and a substantial increase in total extrusion force; moreover, adjusting the layer height can reduce extrusion pressure by up to 60.76%. This force‑prediction model can guide the pre‑setting of slicer parameters, thereby minimizing printing defects and enabling the stable fabrication of high‑precision, thin‑walled silicon components.

II. The Core Carrier Role of Silicone Rubber 3D Printing in This Study

This study relies entirely on a silicone rubber MEX extrusion-based 3D printing system for modeling, simulation, and experimental validation; it serves as the sole physical platform for realizing the model and fulfills four key functions:

• Experimental data acquisition platform, enabling full-scale quantitative in-situ measurements of applied forces.

The in-house-developed silicone rubber extrusion printer has been modified by integrating a high-precision thin-film pressure sensor between the piston and the push plate. This sensor enables real-time measurement of nozzle thrust, interlayer back pressure, and total extrusion force under various conditions—different nozzles (0.84 mm/1.6 mm), varying flow rates, and dual-layer thicknesses—providing authentic reference data for two predictive models and filling the gap in empirically measured force‑related data for silicone printing.

• Model simulation validates the physical benchmark.

Using the rheological parameters of Dow Corning 737 silicone rubber as input for the simulation, a 2D axisymmetric computational domain was established based on the actual print nozzle and the geometric dimensions of the deposited layer. The simulations separately modeled the conical flow field within the nozzle and the backpressure flow field in the bottom deposition layer, and the resulting solid‑state printed line morphology was used to verify whether the extrusion pressure predicted by the model matches the observed deformation defects.

• Process-Regime Verification Platform

Leveraging the layer-by-layer deposition characteristics of 3D printing, we directly demonstrate the strong coupling between layer height and back pressure: a smaller layer height reduces the material flow cross-section, causing a sharp increase in back pressure and a substantial rise in total extrusion force, which in turn leads to filament extrusion and overflow; by contrast, a larger layer height lowers back pressure, resulting in well-defined printed lines, thereby corroborating the mechanical principles derived from the model through empirical print‑out results.

• Platform for Implementing Process Optimization

The optimal combination of nozzle settings, flow rate, and layer height derived from the model can be directly imported into slicing software, enabling the stable fabrication of thin-walled silicone rubber soft structures and medical prosthetics with low porosity and strong interlayer adhesion—without the need for extensive trial prints—and thereby closing the entire closed-loop chain of “simulation‑based prediction–parameter optimization–final‑product manufacturing.”

III. Four Major Core Innovations and Breakthroughs of the Paper

Breakthrough 1. A complete decomposition of the dual‑force system is presented, distinguishing two types of core extrusion loads. Previous studies on silicon‑based printing have focused solely on the thrust within the nozzle, neglecting the interlayer back pressure \(F_C\) exerted by the printing substrate.

This study successfully decomposes the total extrusion force into the nozzle‑fluid thrust \(F_N\) and the reverse back pressure of the deposited layer \(F_C\), establishing a complete mechanical equation \(F_T = F_N + F_C\). By separately modeling each of these two force components, the study addresses the fundamental limitation of previous models—namely, their large prediction errors.

Breakthrough 2. A semi-analytical MATLAB model is developed based on the Jeffery–Hamel (J‑H) flow to analyze non-Newtonian silicone rubber flow within a conical nozzle. The governing equations for steady-state radial J‑H flow in spherical coordinates are solved, and a semi-analytical computational routine is constructed by coupling these with the Carreau rheological constitutive equation. This approach eliminates the need for large-scale simulation software; using only MATLAB, it can rapidly compute nozzle thrust under various nozzle geometries and flow rates. The computational efficiency significantly surpasses that of CFD, while maintaining both accuracy and lightweight implementation.

Breakthrough 3. A pioneering, independent simulation modeling approach for layered deposition under back pressure has been developed, featuring a two-dimensional fluid model that discretizes the control volume (CV) of the bottom-layer deposition. This model simulates the shear flow field induced by nozzle motion and quantifies the influence of layer height and line width on back pressure. It establishes a clear quantitative relationship: lower layer heights, smaller flow cross‑sections, and higher back pressures all contribute to increased back pressure. The method can reduce extrusion pressure by up to 60.76%, providing a quantitative basis for optimizing processes involving thin‑walled components.

Breakthrough 4. High-precision experimental validation of the dual-model approach under multiple operating conditions was conducted using eight sets of comparative experiments—featuring large and small nozzles, high and low flow rates, and double-layer thicknesses—simultaneously validating both the semi-analytical model and the CFD simulation model.

The semi-analytical approach achieves a maximum prediction accuracy of 99.7%, while CFD simulation reaches up to 99.3%. Furthermore, compared with the Newtonian fluid simplified model, it is confirmed that a shear-thinning Carreau fluid must be employed to accurately capture the high-shear conditions in small nozzles, thereby providing a standardized material constitutive model for subsequent silicone rubber printing simulations.

IV. Technical Application Scenarios and Future Development Directions

( 1) Current Real-World Application Scenarios

• Standardization of Soft Robotics R&D Processes

It enables predictive optimization of printing parameters for bionic grippers, artificial muscles, and micro‑scale pipe‑climbing robots, significantly reducing the need for iterative machine adjustments and trial prints, minimizing interlayer porosity and sidewall waviness defects in soft structures, and enhancing the mechanical stability of actuators.

• Manufacture of Medical Silicone Rubber Implants

High-precision medical components, such as meniscal prostheses and silicone scaffolds for soft-tissue repair, are manufactured using model‑based optimization of layer height and extrusion pressure, ensuring that the implants are dense and free of porosity while meeting both biomechanical performance and aseptic processing requirements.

• Flexible Sensing and Microfluidic Chips

Ultra-thin, thin-walled silicone microchannels and flexible sensing substrates, leveraging model-based prediction of extrusion pressure to prevent wall collapse and channel blockage, thereby enhancing the consistency of microstructure fabrication.

• Intelligent Slicing Development for Desktop-Level Silicon 3D Printers

Integrate a semi-analytical model into slicing software; by inputting nozzle parameters and target layer height, the software automatically selects a safe printing speed, enabling one-click optimization of process parameters and lowering the barrier to equipment operation.

(II) Future Directions for Research

• Stress-Extended Modeling of Multilayered Superimposed Structures

The current model addresses only the back pressure at the first deposition layer. Future work will extend it to a multi-layer, continuous‑printing stress model, enabling prediction of the cumulative back pressure arising from multilayer stacking and supporting the printing of tall silicone rubber structures.

• General Model for Multi-Filler-Modified Silicone Ink

The model is adapted to modified inks such as magnetic silicon and conductive composite silicon, incorporating the effects of fillers on rheological parameters, thereby establishing a comprehensive suite of universal force‑prediction tools for silicon‑based printing.

• Machine learning coupled with a semi-analytical model

By training algorithms on vast experimental print‑data sets and correcting model boundary errors, we enable one‑click generation of optimal process windows for any nozzle and filament type, thereby building a fully automated process‑optimization system.

• Electromagnetic-Assisted Printing Composite Mechanical Model

Building on previous research into electromagnetic drag reduction, we incorporate electromagnetic lift into the overall force balance equation and establish a comprehensive predictive framework for electromagnetically assisted silicon printing, tailored to high‑precision fabrication of ultra‑thin layers.

• High-speed continuous printing dynamic force simulation

By incorporating dynamic fluctuations in printing speed and nozzle‑heating variables, the model simulates transient variations in extrusion force under high‑speed mass‑production conditions, thereby supporting industrial‑grade silicone rubber additive manufacturing lines.

Image and Text Guide

Table 1. Silica Gel Parameters

Figure 1. A customized MEX 3D printer designed specifically for silicone extrusion, equipped with a force sensor to measure extrusion force; b) Schematic illustration of the various forces acting within the silicone MEX system.

Figure 2. Modeling of fluid flow inside the nozzle: a) Actual nozzle geometry; b) Two-dimensional axisymmetric model; c) Two-dimensional axisymmetric model used for J‑H modeling (Source: [38]).

Figure 3. a) Internal nozzle and the geometry of the layered structure; b) Top-down view of the deposition process, with the control volume (CV) indicated; c) Three-dimensional model of the computational control volume; d) Boundary conditions.

Table 2. Experimental Nozzle Thrust at Different Flow Rates and Nozzle Sizes

Table 3. Experimental Backpressure under Different Flow Rates, Layer Thicknesses, and Nozzle Sizes

Figure 4. Velocity distribution curves within the second conical control volume of the nozzle: comparison between CFD simulation results and the J‑H semi‑analytical approximation method (dots: MATLAB‑based results; solid line: CFD results); θ: polar angle; Q: flow rate; 𝜙: nozzle exit diameter; vr: radial velocity; θm: position of the maximum azimuthal angle; rk: the kth cross‑section along the control volume (see Figure 9.S‑b).

Figure 5. Velocity distribution curves within the first conical control volume of the nozzle: comparison between CFD results and the J‑H semi‑analytical approximation (dots: MATLAB results; solid line: CFD results); θ: polar angle; Q: flow rate; 𝜙: diameter at the outlet of the first conical section; 𝑣𝑟: radial velocity; θ𝑚: position of the maximum azimuthal angle; rk: the kth cross‑section along the control volume (see Figure 9.S‑b).

Figure 6. Variation of nozzle extrusion force with volumetric flow rate, comparing measured and predicted values; 𝐹𝑁: nozzle extrusion force; 𝑄: flow rate; ∅: nozzle exit diameter.

Table 4. Reduction in Total Force When Switching from Low-Altitude Thickness Values to High-Altitude Thickness Values

Table 5. Accuracy of the CFD and Semi-Analytical MATLAB Models Under All Experimental Conditions

Figure 7. Variation of back pressure with volumetric flow rate and layer thickness, comparing measured and predicted values; 𝐹𝐶: back pressure; Q: flow rate; ∅: nozzle outlet diameter; t: layer thickness.

Figure 8. Variation of the overall extrusion force with volumetric flow rate and layer thickness, as predicted and measured; 𝑭𝑻: overall extrusion force; Q: flow rate; ∅: nozzle exit diameter; t: layer thickness.