Mechanical Characterization and Constitutive Modeling of 3D Printable Soft Materials

Introduction

Unlike traditional engineering designs, soft robots routinely undergo large deformations, ¹ and the mechanical response of their constituent materials is not modeled accurately by a linear stress–strain relationship. Several dozen ² hyperelasticity models of varied mathematical form and complexity have been proposed to describe the underlying behavior of these rubbery materials. In this work, we present hyperelasticity models for 14 popular 3D-printable soft materials, and detail the “recipe” (fabrication, testing, and data processing steps) used to derive them.

The 3D-printable materials suitable for soft robots are rapidly becoming more popular, allowing designers to overcome limitations of traditional fabrication methods and reduce manual assembly steps. While standardization of material models for commonly used castable silicone materials is underway, ³ limited progress has been made in characterizing these 3D-printable materials. Recent efforts underscore the importance of developing a unified database of material models as well as standard practices for experimental material characterization. Marechal et al. investigated the behavior of 17 commercially available nonprintable elastomers and supply raw data as well as model coefficients for common incompressible hyperelasticity models. ³ Azmi et al. highlight the need for standardization in test sample geometry and procedures by comparing ASTM standards in the uniaxial tensile testing of several silicone rubbers, showing 50% disagreement between hyperelastic coefficients fit to data from each standard. ⁴

Bortoli et al. have commercialized a fast and general curve fitting code, Hyperfit, offering several fitting algorithms, 40 hyperelasticity models, and multicriteria optimization. ² Gorrisen et al. aggregate published material models for soft robotic actuators in their broad survey of the field, ¹ and note the lack of standardization in material model selection across the simulation results they review.

In addition to 12 standard thermoplastic polyurethane (TPU) materials, we mechanically characterize two electrically conductive materials, previously tested for their electrical properties. ⁵ These enable the fabrication ⁶ and potential integration ⁷ of robust resistive strain sensors in soft robotic assemblies.

Materials and Methods

Fabrication and test method

Samples were fabricated using a commodity fused filament fabrication 3D printer (Prusa MK3s, Prusa Research) fitted with an upgraded direct drive filament extruder designed for higher torque (Bondtech, AB) and a nickel-coated brass nozzle with 0.6 mm orifice diameter (Bondtech, AB). All samples were printed using identical gcode, generated using the open-source slicing program PrusaSlicer, with 100% infill and linear extrusion rate of 30 mm/s.

To quantify the dimensional accuracy of the fabricated samples, we compare the mass of each sample to that of a theoretical, dimensionally exact sample. Samples exhibit low variability in mass inside each material group, but clear variability across material groups (Fig. 1, right). While softer filaments appear more likely to underextrude, designers can compensate by adjusting the “extrusion multiplier” parameter available in slicing software.

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FIG. 1.

Empirical stress–strain curves for 3D printable soft materials tested in uniaxial tension according to ASTM D412, eight test samples per filament type. Circular marks indicate mean of empirical data, shaded region represents 2σ (95%) confidence bounds, and solid bold lines indicate best-fit third-order Ogden hyperelasticity model. Upper left inset shows infill orientation for samples, which were tested to failure, 550%, or 600% strain depending on material type. Right subplot shows percent error in mass of as-fabricated samples compared with a geometrically perfect sample, displayed in boxplot form (shaded rectangle covers the 25th–75th percentile, horizontal line stretches between the extrema, and vertical line lies at the mean). Softer filaments appear to be more prone to underextrusion, although multiple exceptions to this trend are evident. For color representation of this figure, the reader is referred to the online version of this article.

Test specimens were designed and tested according to ASTM standard D412 (Die C, 33 × 6 × 1.6 mm test region), Test Methods for Vulcanized Rubber and Thermoplastic Elastomers—Tension. ⁸ Samples were stretched to failure or 600% engineering strain, dictated by the maximum travel available on the load frame used for this characterization (810E5 All-Electric Dynamic Test Machine, Test Resources). Testing was performed on eight samples of each material, divided into two groups (A and B), with infill direction-oriented 45° and 90° offset from the pull direction, respectively.

Results

Curve fitting is performed on the mean stretch–true stress response of each material to find values of the Ogden coefficients ( μ i , α i ) , which minimize error between the empirical curve and fit equation. We employ a general nonlinear regression algorithm in the MATLAB (© MathWorks 2022) function fitnlm() to search for optimal Ogden coefficients, and quantify fit quality using the Standard Error of the Estimate S³: S = ∑ i = 1 n ( y i − ŷ i ) 2 n − k − 1(3)

where n is the number of empirical datapoints, y_i is empirical data, ŷ i is predicted (fit) data, and k is the number of predictors (i.e., coefficients to be estimated during fitting). Fitting is performed with Ogden order N = 1 , 2 , 3 for each filament, and Akaike Information Criterion ³ is computed to quantify fit quality relative to model complexity. Optimal Ogden parameters (Table 1) are ready for implementation in any commercial or research (e.g., FeBio ¹⁰ ) numerical analysis code, allowing researchers to analyze soft robot designs without performing their own mechanical testing.

Conclusion

We present a database of 3D printable soft material models, lowering barriers to the wider adoption of simulation soft robotics research. We describe sample fabrication, test procedures, and fitting procedures, adding to earlier work ³ in creating a standardized method for mechanical characterization of materials relevant to soft robotics.

We hope this work spurs adoption of standardized test procedures and hyperelasticity models for common soft robotic materials and shifts focus toward more specialized characterization. In particular, some TPUs tested here exhibit viscoelasticity that falls beyond the scope of this work, but is vital to characterize for soft robotic applications operating in high strain rate contexts. Additionally, further electromechanical characterization of conductive filaments is needed.