Facile Analytical Extraction of the Hyperelastic Constants for the Two-Parameter Mooney–Rivlin Model from Experiments on Soft Polymers

Abstract

Having accurate data to represent hyperelastic materials that underpin soft robotics would facilitate their analysis, design, and validation. We seek to provide the reader with a useful tool to overcome a mundane but crucially important problem in determining the hyperelastic material properties. We show how to employ first dimensionless and then dimensional comparisons between experimental data and the classic theoretical model representing this system to produce C₁ and C₂ for the Mooney–Rivlin model, closely representing a variety of soft polymers.

Introduction

Soft robotics has increasingly gained traction in research and applications over the past decade.¹ This typically involves making actuators of different sizes and configurations using hyperelastic materials, biological materials, or synthetic soft tissues² to perform dextrous maneuvers that would otherwise be impossible to conduct using conventional rigid robots. Hyperelastic materials can be manufactured with biocompatibility and a broad range of elastic moduli, failure strains from 200% to >1000%, surface properties, solvent resistance, and thermal resistance. The material choice depends upon the target application, making it one of the most critical parameters in research design and development. The exponential growth of the soft robotics discipline has been driving a similar growth in the number and variety of hyperelastic materials. Understanding and controlling their behavior in applications are crucial to continue this growth.

To design and characterize the behavior of hyperelastic polymer materials, various constitutive models have been developed in the past. Beyond the classic neo-Hookean model,³ the theory for large deformations of hyperelastic polymers was initiated by the seminal work of Mooney⁴ and further elaborated by Rivlin and Saunders,⁵ which has come to define the Mooney–Rivlin model. Other well-known models include the Ogden,⁶ Yeoh,⁷ Blatz-Ko,⁸ Arruda-Boyce,⁹ and Gent¹⁰ models. Each of these constitutive models requires one or more hyperelastic constants, and finding values for those constants typically requires a regimen of experiments.

In the case of the neo-Hookean model, uniaxial tests are sufficient to fully determine the single constant that it requires to define the material. However, most other models require more than one material constant, and so more experimental tests are required, usually among the following choices: pure shear, equibiaxial tension, planar tension, uniaxial compression, uniaxial tension, and biaxial membrane (bulge) tests.¹¹ An alternative approach using atomic force microscopy nanoindentation tests was studied by Dimitriadis et al.,¹² and subsequently in a few more studies^13–15 wherein parameters B₁ and B₂ represent the hyperelastic constants. This approach requires extraction of B₁ and B₂ using Young's modulus E, however, there is no established relationship between B₁, B₂, C₁, and C₂. There has been demonstration of a purely computational approach, without experimental validation, that involves a finite element model (using ABAQUS, Dassault Systèmes, Vélizy-Villacoublay, France) comparing the results obtained from neo-Hookean, Mooney–Rivlin, and Yeoh models.

Other approaches involve extraction of the storage modulus^13,16 to iteratively tailor the behavior of polydimethylsiloxane (PDMS). The tensile and compressive moduli can also be extracted using¹⁷ tensile and compressive testing.

In the two-parameter Mooney–Rivlin model, second-order Yeoh, and second-order Ogden models, two hyperelastic constants are required. For instance, the first and second hyperelastic constants C₁ and C₂, also referred to as $C_{10}$ and $C_{01}$, respectively, in the two-parameter Mooney–Rivlin case. Although uniaxial tensile tests are sufficient to obtain the first hyperelastic constant C₁, biaxial membrane tests are typically necessary to obtain the second hyperelastic constant C₂. Various methods and results have been presented to extract the hyperelastic constants^18,19; however, there are limited benchmark results for the values of hyperelastic constants of common materials such as PDMS (Sylgard™ 184; Dow Corning Corp., Midland, MI) and many other proprietary hyperelastic materials, such as the platinum-cure silicone Dragon Skin (Smooth-On Corp., Macungie, PA).

The first reported values for PDMS were $C_1=34.3$ and $C_2=46.9$ kPa,²⁰ obtained from uniaxial tensile tests. In a subsequent study, uniaxial experiments were conducted in both axisymmetric and asymmetric manner, and computational simulations using constitutive models were used to compare against experimental data and establish material constant values for the models.²¹ Sasso et al.¹¹ added biaxial tests to uniaxial tests on rubber-like materials and validated the results with finite element analysis (ABAQUS) to determine the values of the material constants for the neo-Hookean, Arruda-Boyce, Mooney–Rivlin, Ogden, and Yeoh material models.

Later, the results of ultralarge deformation bulge tests conducted on PDMS microballoons were compared with those of finite element analysis (ANSYS, Inc., Canonsburg, PA) to obtain $C_1=75.5$ and $C_2=5.7$ kPa for PDMS,²² remarkably different than Kawamura et al.²⁰ Uniaxial tensile and compression tests of dielectric elastomers have been used in conjunction with finite element analysis (ABAQUS) to determine the properties of dielectric elastomers,²³ although these materials would certainly be expected to exhibit mechanical properties different than pure PDMS.

The overall approach in the literature has been to validate experimentally obtained results with finite element analysis, and then use the results of this validation to derive the appropriate constants necessary to represent the material with an analytical model. Although the use of computation to compare with experimental results and determine the constants for use in an analytically derived model is convenient, it is tedious and subject to modeling and representation errors made in finite element analysis.

Since it is actually possible to directly use the theory in determining the appropriate values for representative constants such as C₁ and C₂—without having to resort to computational analysis—a viable alternative is to omit the computations. This avoids potential problems arising from approximate representation of the boundary conditions and interfaces present in the system when using finite element analysis. The theoretical approach likewise eliminates the computational costs associated with obtaining and running finite element analysis models.

We consider as an example of this approach the inflation of a small cylindrical disk of hyperelastic media restrained at the periphery. Christensen and Feng²⁴ long ago combined the theory developed by Mooney⁴ and Rivlin²⁵ with an approximate closed-form solution for the inflation of a thin circular disk to determine the hyperelastic material constants, but there has been no demonstration since to use this theory to produce, for example, the C₁ and C₂ values for PDMS. In other words, no one has used a purely analytical approach to validate the experimental data. However, over this same time many^21,11,26 have used the comparatively painstaking approach of introducing computations to determine these values, as they enable analogous representations of the respective experiments conducted on the hyperelastic materials. In this article, a nondimensionalized analytical validation method is provided as a universally applicable approach regardless of the details, provided the minimum criterion for the stretch ratio is met as discussed later.

Methods

Experiments

Different hyperelastic materials will exhibit independent hyperelastic constants. Uniaxial tensile tests are frequently employed to extract the first hyperelastic constant C₁; however, to obtain both C₁ and C₂, a biaxial membrane test is required. Biaxial tests can be conducted in different configurations, either beginning with a three-dimensional balloon or biaxially inflating an initially flat disk. Making uniform flat thin films is easier than fabricating three-dimensional balloons with good surface uniformity out of unknown materials, and so we adopt the biaxially inflated flat disk approach.

Thin films of five different hyperelastic materials, PDMS (Sylgard 184), Dragon-Skin™ 10, Ecoflex™ 30, Sorta-clear™ 40, and Dragon-Skin mixed with hexane (weight ratio Dragon-Skin A:B:hexane = 1:1:1), were made by combining the base and curing agent in the manufacturer recommended ratio. Glass slides were coated with release agent (Ease Release™ 200; Mann Release Technologies, Macungie, PA). All the hyperelastic polymers were spin-coated at 750 rpm for 60 s. Upon curing, the resulting thin films were placed in a custom fixture shown in Figure 1.

FIG. 1.

(a) A Luer connector base and steel plate with hole provided to pass the membrane through when inflated, shown (b) from slightly above the horizontal view. The membrane, upon hydraulic inflation from the Luer lock expands as a nearly spherical shape to produce a vertical height above the steel plate, Δ that grows with inflation until membrane failure. The O-ring beneath clamps the membrane in place against the steel plate.

The 3 mm diameter of the circular hole in the base plate and steel plate is similar to eliminate potential lateral stretching. The 4 mm diameter, 0.5 mm thickness O-ring prevents air leakage between the Luer connector base and the steel plate; it is sized to be commensurate with the inner diameter of the steel plate to prevent interference with the membrane expansion from the circular hole. The entire assembly ensures the entire membrane's edge along a circular section is clamped together, leaving a 3 mm radius circular region free to inflate in the middle from an inlet in the substrate to which the membrane and clamp were fastened. Water was introduced through a Luer lock connection into the chamber formed by the clamped membrane and inlet using a high-precision microfluidic system (microfluidics control, OB1; Elveflow^®, Paris, France). Water was used as the driving fluid to eliminate the potential effects of compressibility associated with pneumatic (air or gas) inflation. The membranes were inflated to failure with pressure increments of 3 kPa/s and filmed using a high-speed camera (Fastcam Mini UX100; Photron, Irvine, CA) combined with a long-distance microscope (CF–1; Infinity, Boulder, CO). The Elveflow system provided the inflation pressure as a function of time to a computer, and was used to trigger the camera to synchronize inflation of the membrane with video recording saved on the same computer. The deformation of the membranes as a function of the inflation pressure was obtained using custom image processing code on MATLAB (MATLAB, Natick, MA).

Analysis

To evaluate the hyperelastic constant values, a relation between the inflation pressure and deformation of the membrane must be established. For a hyperelastic material, the Mooney–Rivlin form of the strain energy function, W, is defined as^4,5

$$W=C_1\left(J_1-3\right)+C_2\left(J_2-3\right).$$ (1)

The constant C₁ is proportional to the number of molecular strands per unit volume and C₂ represents additional restraints on the molecular strands; the shear modulus $G=2\left(C_1+C_2\right)$ while J₁ is a measure of the strain defined as²⁷

$$J_1={\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2,$$ (2)

and J₂ is defined as

$$J_2={\lambda }_1^{-2}+{\lambda }_2^{-2}+{\lambda }_3^{-2}.$$ (3)

When an initially flat circular disk of an elastomeric material with radius R is inflated in a biaxial membrane, or “bulge,” test to produce a spherical cap shape, the stretch ratio, λ, for the maximum displacement of the membrane, Δ, is defined as²⁴

$$\lambda =\left(\frac{\frac{\Delta }{R}+\frac{R}{\Delta }}{2}\right){sin}^{-1}\left(\frac{2}{\frac{\Delta }{R}+\frac{R}{\Delta }}\right).$$ (4)

The relationship between the inflating pressure, P, the maximum deformation of the membrane, Δ, and the stretch ratio, λ, is defined as²⁴

$$P=\frac{8C_{1}H}{R\left(\frac{\Delta }{R}+\frac{R}{\Delta }\right)}\left[\left(1-\frac{1}{{\lambda }^6}\right)+\alpha \left({\lambda }^2-\frac{1}{{\lambda }^4}\right)\right],$$ (5)

where H is the thickness of the membrane and $\alpha =C_2∕C_1$ is the ratio of the hyperelastic constants.

Nondimensionalizing the pressure and the deformation as

$$\tilde{P}=\frac{RP}{4C_{1}H}and\tilde{\Delta }=\frac{\Delta }{R_0},$$

where R₀ is the initial radius of the membrane, produces

$$\tilde{P}=\frac{2}{\left(\tilde{\Delta }+\frac{1}{\tilde{\Delta }}\right)}\left[\left(1-\frac{1}{{\lambda }^6}\right)+\alpha \left({\lambda }^2-\frac{1}{{\lambda }^4}\right)\right],$$ (6)

where

$$\lambda =\left\{\begin{array}{ccc}\frac{\left(\tilde{\Delta }+\frac{1}{\tilde{\Delta }}\right)}{2}{sin}^{-1}\left(\frac{2}{\frac{\tilde{\Delta }}{R}+\frac{R}{\tilde{\Delta }}}\right)& if& \tilde{\Delta }\le 1\\ \frac{\left(\tilde{\Delta }+\frac{1}{\tilde{\Delta }}\right)}{2}\left(\pi -{sin}^{-1}\left(\frac{2}{\frac{\tilde{\Delta }}{R}+\frac{R}{\tilde{\Delta }}}\right)\right)& if& \tilde{\Delta }\ge 1\\ \end{array}\right..$$ (7)

The dimensionless analytical pressure $\tilde{\text{P}}$ was obtained as a function of the stretch ratio $\tilde{\Delta }$ using Equation (6). The experimental $P_{exp}$ was scaled by $P_R\equiv {\left.P_{exp}\right|}_{\Delta =R}$ to produce the nondimensional pressure ${\tilde{P}}_{exp}=P_{exp}∕P_R$. The pressure P_R is the pressure required for the membrane to deform an amount equivalent to the initial membrane radius, $\Delta =R$. For an increase dr of the membrane radius, the expansion of the surface requires an amount of work proportional to the pressure. The hyperelastic constants define the factor of proportionality.

Least-squares minimization was used to determine the best value of α in minimizing the error between the dimensionless experimental pressure–deformation data and the analytical dimensionless pressure obtained from Equation (6) across the stretch ratios. Since the materials tested in this study have different mechanical properties, the values of P_R and α are different for each material.

After establishing the value of α, the dimensional pressure _P was evaluated in terms of the dimensional deformation Δ using Equation (5), where the thickness of the membranes was obtained using a surface profilometer (Dektak 150; Veeco, Plainview, NY). Least-squares minimization was performed to determine the value of C₁ that minimizes the error between the dimensional experimental pressure–deformation data and the analytical pressure–deformation obtained from Equation (1). Since the value of α is known from the previous step, $C_2=C_1\alpha$ can also be determined.

Results

The experimental nondimensionalized pressure is plotted with respect to the stretch for two of the materials, Ecoflex and Sylgard 184 (PDMS), as shown in Figure 1. The remaining plots of the materials are provided in the Supplementary Data.

Least squares minimization was used between the experimental data and Equation (3) to determine the value of α and obtain the best-fit plots in Figure 2.

FIG. 2.

Dimensionless inflation pressure, $P∕P_R$ plotted with respect to the dimensionless deformation of the membrane, $\Delta ∕R_0$ (x-axis) for (a) Ecoflex™ and (b) Sylgard™ 184 (PDMS). Least-squares minimization was used to determine the value of the ratio of the hyperelastic constants, α, to directly fit the nondimensionalized analytical model to the nondimensionalized experimental data (Ecoflex: $\alpha =0.1$, $R^2=0.91$; PDMS: $\alpha =0.04$, $R^2=0.99$). PDMS, polydimethylsiloxane.

Once the value of ratio of the hyperelastic constants, α, is found, the value of C₁ may then be found by least squares fitting of Equation (1) to the dimensional pressure–deformation results as shown in Figure 3.

FIG. 3.

Dimensional inflation pressure, P, as a function of the maximum deformation of the membrane, Δ for (a) Ecoflex™ and (b) Sylgard™ 184 (PDMS). Least squares minimization was used to find the best fit for C₁ knowing the appropriate value of α from the dimensionless fitting. The values of C₁ found from this procedure are provided in Table 1 and $R^2=0.89eqnoopen\left(Ecoflex\right)$ and $R^2=0.99eqnoopen\left(PDMS\right)$.

Table 1 lists all the materials tested in this study, and their respective values of C₁ and C₂, indicating an ability to test a rather broad range of materials, and likewise notably indicating the remarkable range possible from relatively similar soft polymer materials.

Table 1.

Values of Hyperelastic Constants C₁ and $C_2=\alpha C_1$ Obtained by Fitting the Two-Parameter Mooney–Rivlin Model

Material	C1 (kPa)	C2 (kPa)
Dragon-Skin™ 10	180	11.7
Ecoflex™	75	7.5
Sorta-Clear™ 40	830	52.3
Sylgard™ 184 (PDMS)	270	10.8
Dragon-Skin 10+Hexane at 1:1 weight ratio	180	14.9

PDMS, polydimethylsiloxane.

Table 2 shows the values of the hyperelastic constants C₁ and C₂ obtained for PDMS compared with previously published results.

Table 2.

Comparison of Hyperelastic Constant Values for Sylgard 184 (Polydimethylsiloxane) Through the Analytical Approach and Previous Studies with Biaxial Membrane (Bulge) Tests

Sylgard™ 184 (PDMS)	C1 (kPa)	C2 (kPa)
Analytical approach	270.0	10.8
Kawamura et al.20	34.3	46.9
Yoon et al.22	75.5	5.7

Conclusions

Biaxial membrane tests were conducted on five different hyperelastic materials. High-resolution imagery, high-precision microfluidic systems, and custom MATLAB image processing code were used to extract the inflation pressure as a function of the stretch ratio for the hyperelastic membranes. Each of the membranes was tested to failure to ensure the minimum criterion for the stretch ratio was met.

As an example, the two-parameter Mooney–Rivlin model was used in conjunction with an analytical approximation for the large deformation of elastic disks to establish the hyperelastic material properties of five different hyperelastic materials. This approach, although quite simplified, is applicable to a large variety of hyperelastic materials enabling the extraction of their hyperelastic constants through analytical validation of the experimental data. Although the values of the hyperelastic constants in previous studies substantially differ between themselves and the values we found, this can be attributed to different material formulations¹⁶ and curing conditions, particularly temperature and cure duration.¹⁷

However, the facile method presented in this article provides a means to easily determine the mechanical characteristics of a hyperelastic material. The method avoids computational analysis that introduces problems of its own, and can be quickly performed so that many material choices can be considered in design. Because it is simple to set up and conduct, the method also enables the reader to quickly produce useful property values from their own materials that may be then used in more accurately modeling their robots and other devices.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

Funding for this work was gratefully provided by the American Heart Association's Innovative Project Award (19IPLOI34760705), the state of California's AB2664 Medical Entrepreneurship Education and Training Grant scheme, UCSD's Galvanizing Engineering in Medicine program, and the National Institutes of Health via the University of California Center for Accelerated Innovation (NIH/NCATS UCSD CTRI 1UL1TR001442-01).

Supplementary Material

Supplementary Data

Supplementary Figure S1

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Supplementary Figure S4

Supplementary Figure S5

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Supplementary Figure S7

Supplementary Figure S8

Supplementary Figure S9

Supplementary Table S1