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. 2026 Sep 28;11(40):60797–60807. doi: 10.1021/acsomega.6c03650

Small-Strain Tensile Characterization and Finite Element Calibration of Sodium Alginate-Based Biomimetic Artificial Muscle Strips

Xuan Wen †, Lan Wang †, Yan Xu †,*, Yanzhuo Lv ‡, Mingao Lei ‡, Chuan Liu ‡
PMCID: PMC13670803  PMID: 42889984

Abstract

This study investigates the small-strain tensile response and finite element calibration of sodium alginate-based biomimetic artificial muscle strips and extends the calibrated mechanics to an electro-ionic-mechanical model. The Ni-doped formulation was selected from our previous optimization of actuation and electrochemical performance. Quasi-static uniaxial tensile tests were performed on three undoped and three Ni-doped specimens. Equivalent moduli were extracted over 0–5% engineering strain, where both groups showed approximately linear responses. The undoped and Ni-doped groups exhibited moduli of 0.975 ± 0.212 MPa and 1.118 ± 0.474 MPa. With n = 3 per group, the between-group comparison was inconclusive (Welch’s t test, p ≈ 0.67) and should not be interpreted as evidence of equivalence. A two-dimensional plane-stress model was calibrated against the group-level tensile responses. The model reproduced the initial stiffness and showed stress concentration near the fixed end and interlayer transition regions. The calibrated mechanical framework was extended under 3 V excitation by coupling electrostatics, ion transport, differential swelling, and solid mechanics. The multiphysics model reproduced the overall 0–1000 s output-force evolution with an RMSE of 0.0992 mN, an MAE of 0.0757 mN, R2 = 0.8152, and a plateau-force error of 3.76%. These results provide experimentally based mechanical parameters and a computational framework for subsequent voltage-dependent studies.

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1. Introduction

Hydrogels demonstrate extensive application potential in fields such as biomimetic artificial muscles, flexible sensors, and tissue engineering, attributed to their high-water content, exceptional flexibility, and favorable biocompatibility. − Among various hydrogel systems, sodium alginate-based hydrogels have emerged as one of the pivotal matrix materials in biomimetic artificial muscle research due to their abundant availability, facile cross-linking capability, excellent film-forming properties, and compatibility with materials such as graphene, carbon nanotubes, poly­(vinyl alcohol), and polyacrylamide to form composite networks. ,− However, sodium alginate-based biomimetic artificial muscles commonly encounter challenges, including low elastic modulus, limited tensile strength, and inadequate mechanical stability, which further restrict their engineering applications in terms of actuation performance, structural reliability, and operational lifespan. − For hydrogel-based artificial muscles, reliable mechanical parameters are essential not only for structural evaluation but also for subsequent electromechanical multiphysics modeling.

To improve the mechanical and functional performance of sodium alginate-based hydrogel actuators, various modification strategies have been reported, including conductive filler incorporation, double network construction, fiber reinforcement, and multiple cross-linking design. Previous studies have shown that GO or PSS incorporation can improve the electromechanical or chemical properties of alginate artificial muscles, while excessive filler content may adversely affect stiffness. Interpenetrating-network and double-network strategies have also been used to regulate modulus, stress relaxation, and fracture resistance of hydrogel materials. ,− In addition, fiber-reinforced and coordination-assisted hydrogel systems have demonstrated improved tensile strength, elastic modulus, and mechanical durability. − These studies indicate that the mechanical response of hydrogel-based artificial muscles is strongly affected by material composition, network structure, and interfacial state.

In our previous work, Ni incorporation into the cross-linking system of sodium alginate-based biomimetic artificial muscles was optimized with respect to actuation and electrochemical performance. , The present study used the previously optimized Ni-doped formulation. The focus is on its small-strain tensile response and comparison with the undoped material.

Mechanical characterization is essential for evaluating hydrogel-based artificial muscles and for providing material parameters for finite element modeling. Uniaxial tensile testing is commonly used to obtain stress–strain response, equivalent modulus, and deformation characteristics, while dynamic mechanical analysis can provide Supporting Information regarding testing-protocol-dependent mechanical response. Previous studies have described the nonlinear, viscoelastic, and structure-sensitive behavior of soft hydrogels using tensile tests, constitutive fitting, and numerical modeling approaches such as piecewise linear, Ogden, Marlow, and other hyperelastic or viscoelastic models. − These studies suggest that reliable mechanical parameter identification is a necessary basis before more complex electromechanical or multiphysics models are constructed. ,

Finite element modeling is widely used to analyze deformation and stress distributions in hydrogel structures. Previous studies have addressed swelling deformation, thermoresponsive or pH-responsive bending, humidity-driven instability, and shape-programmable hydrogel actuators. − Mechanical parameters for layered sodium alginate-based artificial-muscle strips remain sample-specific and require experimental identification. A small-strain mechanical calibration is needed before electric-field effects, ion transport, swelling deformation, or time-dependent internal variables are introduced. ,

This work focuses on small-strain tensile characterization and finite-element calibration of sodium alginate-based biomimetic artificial muscle strips. The objectives are to (i) determine quasi-static tensile parameters of undoped and previously optimized Ni-doped specimens within 0–5% engineering strain; (ii) quantify group-level differences while retaining specimen-to-specimen variability; (iii) calibrate a two-dimensional plane-stress model for the small-strain tensile response; (iv) evaluate mesh convergence and local stress transfer in the layered structure; and (v) extend the calibrated mechanical parameters to an electro-ionic-mechanical model under 3 V excitation.

2. Experimental Procedures and Finite Element Modeling

2.1. Specimen Geometry and Layered Structure

All specimens were cut into rectangular strips 30 mm long and 5 mm wide. The total thickness was approximately 0.414 mm. The base film was approximately 0.30 mm thick, each electrode film was approximately 0.05 mm thick, and each cross-linked gel layer was approximately 0.007 mm thick. The through-thickness sequence was electrode film/cross-linked gel/base film/cross-linked gel/electrode film, forming a five-layer sandwich structure (Figure ). The Ni-doped specimens were prepared by using the previously optimized Ni-doping ratio.

1.

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Geometry and layered architecture of the sodium alginate-based biomimetic artificial muscle strip: (a) top-view photograph of a representative specimen; (b) side-view photograph of the strip; (c) three-dimensional schematic of the layered structure, including the electrode films, cross-linked gel layers, and base film.

2.2. Selection of Ni-Doped Formulation

The Ni-doped formulation was selected from our previous optimization study of sodium alginate-based biomimetic artificial muscles. , A Ni-doping ratio of 0.25 provided the best combined actuation and electrochemical performance, with a maximum stable output force of 2.405 mN, a bending displacement of 9.61 mm, a specific capacitance of 1100.85 mF·g–1, and an internal resistance of 11.14 Ω. This formulation was used as the Ni-doped representative group for tensile characterization and finite element calibration.

2.3. Experimental Setup and Tensile Testing Conditions

Quasi-static tensile tests were conducted by using an electronic universal material testing machine (INSTRON-3365, Instron, USA) under displacement-controlled loading. Before testing, the artificial muscle films were cut into rectangular-strip specimens. The nominal specimen size was approximately 30 mm in length and 5 mm in width with an overall thickness of approximately 0.414 mm. During testing, both ends of each specimen were clamped by the grips of the testing machine, and uniaxial tensile loading was applied at a constant crosshead speed of 10 mm/min. The specific data are shown in Table .

1. Quasi-Static Tensile Testing Conditions.

Parameter Value/Description
Testing instrument INSTRON-3365 electronic universal material testing machine
Loading mode Quasi-static uniaxial tensile loading
Control mode Displacement control
Loading rate 10 mm/min
Specimen shape Rectangular strip
Specimen length 30 mm
Specimen width 5 mm
Overall specimen thickness 0.414 mm
Base film thickness 0.30 mm
Electrode film thickness 0.05 mm on each side
Cross-linked gel layer thickness 0.007 mm on each side
Initial gauge length, L 0 Recorded individually for each specimen from the original testing file
Number of specimens 3 undoped specimens and 3 Ni-doped specimens
Number of tensile tests One tensile test for each independent specimen
Testing temperature Room temperature
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The effective gauge length depends on the specimen cutting length, grip position, and clamping depth. Engineering strain was calculated using the specimen-specific initial gauge length, L 0, recorded in each original tensile testing file.

As shown in Table , three independently prepared specimens were tested for each group, namely SA-1, SA-2, and SA-3 for the undoped group and Ni-1, Ni-2, and Ni-3 for the Ni-doped group. Each independent specimen was tested once under the same tensile loading protocol. The group mean and standard deviation of the equivalent modulus were calculated from the three independent specimens in each group.

2. Specimen-Specific Gauge Length Used for Engineering Strain Calculation.

Group Specimen Actual gauge length, L 0/mm
Undoped SA-1 14.66
  SA-2 20.76
  SA-3 12.14
Ni-doped Ni-1 17.20
  Ni-2 17.22
  Ni-3 17.29
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2.4. Engineering Stress–Strain Calculation and Modulus Extraction

The tensile testing system recorded the force–displacement data in real time during uniaxial stretching. For each specimen, the actual initial gauge length, L 0, recorded in the original testing file was used to calculate the engineering strain.

Engineering strain (Equations ) was calculated as

εeng=ΔLL0 1

where ε eng is the engineering strain, ΔL is the displacement increment recorded during tensile loading, and L 0 is the actual initial gauge length of each specimen.

Engineering stress (Equations ) was calculated as

σeng=FA0,⁣A0=w×h 2

where σeng is the engineering stress, F is the tensile force, A 0 is the initial cross-sectional area, w is the specimen width, and h is the initial specimen thickness. In this study, the nominal specimen width was 5 mm and the overall specimen thickness was approximately 0.414 mm. Therefore, the nominal initial cross-sectional area was approximately: A 0 = 5 mm × 0.414 mm = 2.07 mm2.

Because the stress–strain response of the specimens was approximately linear in the small-strain region, the 0–5% engineering strain range was selected for equivalent modulus extraction. Linear regression (Equation ) was performed using the following expression:

σeng=Eeq×εeng+b 3

where E eq is the equivalent small-strain Young’s modulus and b is the stress intercept. The intercept was fitted freely rather than constrained to zero.

2.5. Dynamic Mechanical Analysis as Supplementary Characterization

Dynamic mechanical analysis (DMA) was performed as a supplementary mechanical characterization method using a TA Instruments Q800 dynamic mechanical analyzer. The test was conducted under the “DMA Controlled Force” mode with a stress/strain method. The specimens were tested using a film tension fixture, and the specimen dimensions were recorded individually in the original DMA files. The Poisson’s ratio used in the DMA analysis file was 0.440. The specific data are shown in Table . The relaxation modulus recorded by the DMA software was used to compare the apparent stiffness of different samples under the same controlled-force stress/strain testing protocol. The quasi-static tensile data were used for finite element calibration, whereas the DMA data were used for protocol-dependent mechanical comparison. The mechanical finite element model used a Poisson’s ratio of 0.49, as listed in Table S2 of the Supporting Information.

3. DMA Testing Conditions and Recorded Variables.

Parameter Value/Description
Testing instrument TA Instruments Q800 dynamic mechanical analyzer
Software TA Instruments Thermal Analysis, Universal V4.5A
Test mode DMA Controlled Force
Test method Stress/Strain
Fixture Film tension fixture
Specimen shape Rectangular strip
Specimen dimensions Recorded individually for each specimen
Testing temperature Approximately 29 °C
Poisson’s ratio 0.440
Recorded variables Time, temperature, strain, stress, relaxation modulus
Use of DMA data Protocol-dependent mechanical comparison
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2.6. Statistical Analysis

The equivalent moduli are reported as mean ± standard deviation. With three specimens per group and unequal variances, Welch’s t test was used to compare the undoped and Ni-doped groups. A p value lower than 0.05 was considered statistically significant.

2.7. Finite Element Model Development

A two-dimensional finite-element model was developed in COMSOL Multiphysics 6.3. The specimen was represented as a long, thin-layered strip with no out-of-plane constraint during tensile loading. A plane stress formulation was used. The out-of-plane width was set to 5 mm, the specimen length was assigned to the x direction, and the thickness was assigned to the y direction. The displacement field was discretized by using quadratic serendipity shape functions. This two-dimensional representation retains the dominant tensile response while reducing the computational cost of the parameter calibration and parametric analysis.

2.8. Material Parameters and Boundary Conditions

The layered geometry retained the base film, two electrode films, and two cross-linked gel layers. The model length was 30 mm, the out-of-plane width was 5 mm, and the total thickness was approximately 0.414 mm.

All domains were modeled as time-independent linear elastic materials within the 0–5% engineering-strain calibration range. The final effective Young’s moduli were 0.650 MPa for the base film and 0.780 MPa for the electrode films. The effective cross-linked-gel modulus was 0.520 MPa in the undoped model and 3.520 MPa in the Ni-doped model. A Poisson’s ratio of 0.49 was assigned to all mechanical domains. The complete COMSOL parameter definitions are provided in Table S2 of the Supporting Information.

The left end was fixed in both displacement components. A prescribed axial displacement was applied to the right end, while transverse displacement remained free. The maximum axial displacement was 1.5 mm, corresponding to 5% engineering strain for a 30 mm model length. Finite element engineering strain (Equation ) and stress (Equation ) were calculated as

εFE=uL 4

where ε FE is the finite element engineering strain, u is the applied displacement, and L is the model length.

σFE=FrA0,⁣A0=W×ttotal 5

where σFE is the finite element engineering stress, F r is the reaction force at the loaded end, W is the out-of-plane width, and t total is the total specimen thickness. The calculated finite element stress–strain curves were then compared with the experimental curves for parameter calibration.

2.9. Mesh Sensitivity Analysis

A mesh convergence study was performed using the undoped model at 5% engineering strain. The layered geometry was discretized with a mapped quadrilateral mesh. A graded distribution was applied along the specimen length, with finer elements near the fixed end and coarser elements toward the loaded end. Coarse, medium, and fine meshes contained 1200, 4800, and 19200 elements. The displacement interpolation remained quadratic serendipity for all cases. Mesh convergence was assessed using three quantities: the global engineering stress, the maximum local von Mises stress, and the maximum local axial strain. The local quantities were evaluated along a through-thickness cut line at x = 0.5 mm from the fixed end. Relative differences (Equations ) were calculated with the fine mesh as the reference. Detailed mesh division parameters are provided in Table S4 of the Supporting Information.

Difference(%)=|Qmesh−Qfine|/|Qfine|×100% 6

2.10. Electro-Ionic-Mechanical Multiphysics Model

The mechanically calibrated model was extended in COMSOL Multiphysics 6.3 to couple electrostatics, two-species ion transport, concentration-difference-driven differential swelling, and solid mechanics under 3 V excitation. The positive-ion and negative-ion concentrations are denoted by c p and c n , and dc = c p – c n was used as the ionic driving variable. A sequential solution procedure was applied over 0–1000s. The output force was evaluated from the axial reaction force at its right end. The complete electrostatic, ion-transport, and time-dependent actuation parameters are provided in Tables S5 and S6 of the Supporting Information.

3. Results and Discussion

3.1. Quasi-Static Tensile Response and Equivalent Modulus Extraction

The quasi-static tensile response of the sodium alginate-based artificial muscle strips was analyzed to obtain the equivalent small-strain modulus required for finite element calibration. Since the present work focuses on mechanical parameter identification for subsequent multiphysics modeling, the modulus was extracted from the 0–5% engineering strain range, where the stress–strain curves showed approximately linear behavior. For each specimen, the actual gauge length recorded in the original tensile testing file was used to calculate engineering strain. Each independent specimen was tested once, and the group-level results were calculated from three independent specimens.

3.1.1. Undoped Samples

Figure shows the tensile stress–strain response and linear fitting results of the undoped specimens. In the 0–5% engineering strain range, SA-1, SA-2, and SA-3 exhibited approximately linear stress–strain behavior, indicating that the initial tensile response can be described by using an equivalent small-strain modulus. The fitted moduli of SA-1, SA-2, and SA-3 were 0.9238, 0.79376, and 1.2082 MPa, respectively. The mean modulus of the undoped group was 0.975 MPa with a standard deviation of 0.212 MPa.

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Quasi-static tensile response and small-strain linear fitting results of undoped sodium alginate-based biomimetic artificial muscle strips: (a) overall engineering stress–strain curves of SA-1, SA-2, and SA-3; (b-d) linear fitting results of SA-1, SA-2, and SA-3 within the 0–5% engineering strain range, respectively. The shaded regions represent the 95% confidence bounds of the linear fits.

The relatively low modulus indicates that the undoped sodium-alginate-based artificial muscle strips maintained good compliance, which is consistent with the soft nature of hydrogel-based actuator materials. The specimen-to-specimen variation may be related to differences in film thickness, clamping conditions, local microstructure, and specimen preparation.

3.1.2. Ni-Doped Samples

Figure presents the tensile stress–strain response and 0–5% linear fitting results of the Ni-doped specimens. Similar to the undoped group, the Ni-doped specimens also showed an approximately linear behavior in the small-strain range. The fitted moduli of Ni-1, Ni-2, and Ni-3 were 1.42605, 1.35502, and 0.57178 MPa. The mean modulus of the Ni-doped group was 1.118 MPa, with a standard deviation of 0.474 MPa.

3.

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Quasi-static tensile response and small-strain linear fitting results of Ni-doped sodium alginate-based biomimetic artificial muscle strips: (a) overall engineering stress–strain curves of Ni-1, Ni-2, and Ni-3; (b-d) linear fitting results of Ni-1, Ni-2, and Ni-3 within the 0–5% engineering strain range, respectively. The shaded regions represent the 95% confidence bounds of the linear fits.

Although the Ni-doped group had a numerically higher mean equivalent modulus, it also exhibited substantial within-group variability (SD/mean = 42.4%). Ni-3 yielded an equivalent modulus of 0.57178 MPa, compared with 1.42605 MPa for Ni-1 and 1.35502 MPa for Ni-2. The Ni-1/Ni-3 and Ni-2/Ni-3 ratios were approximately 2.49 and 2.37. The lower value of Ni-3 may reflect local inhomogeneity in Ni incorporation or cross-linking, specimen-specific gauge-length uncertainty, or clamping differences. Because only three independent specimens were available, there was insufficient evidence to exclude Ni-3 as a statistical outlier; it was retained in the group mean, standard deviation, and Welch’s t test.

Several fitted lines showed nonzero stress intercepts. The intercepts were −8.57 × 10–4, + 0.00486, + 0.00325, −0.0102, + 0.02267, and −0.00170 MPa for SA-1, SA-2, SA-3, Ni-1, Ni-2, and Ni-3. In particular, the +0.02267 MPa intercept of Ni-2 is nontrivial relative to the stress measured within the 0–5% strain interval. These offsets may arise from residual preload, force-baseline offsets, grip seating, or uncertainty in identifying the zero-displacement and gauge-length references. The fitted slopes are interpreted as equivalent small-strain moduli for the 0–5% interval rather than intrinsic Young’s moduli referenced to an ideal zero-stress state.

3.1.3. Statistical Comparison Between Groups

Welch’s t test was used to compare the equivalent moduli of the two groups. The undoped group had a modulus of 0.975 ± 0.212 MPa, and the Ni-doped group had a modulus of 1.118 ± 0.474 MPa (Table ). The Ni-doped group showed a higher mean value, but the difference was not statistically significant (p ≈ 0.67; Figure ).

4. Equivalent Small-Strain Modulus and Statistical Comparison of Undoped and Ni-Doped Specimens.
Group Specimen E eq /MPa Mean ± SD/MPa p-value
Undoped SA-1 0.9238 0.975 ± 0.212
  SA-2 0.79376    
  SA-3 1.2082    
Ni-doped Ni-1 1.42605 1.118 ± 0.474 0.67
  Ni-2 1.35502    
  Ni-3 0.57178    
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a

E eq was obtained from linear fitting within the 0–5% engineering strain range. The p-value was obtained using Welch’s t test (p = 0.67).

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Statistical comparison of the equivalent small-strain modulus between the undoped and Ni-doped groups. Individual specimen values are shown together with the mean ± standard deviation. Statistical comparison was performed using Welch’s t test (p = 0.67).

3.2. Supplementary DMA Results Under Controlled-Force Stress/Strain Testing

DMA was used as a supplementary characterization method to compare the apparent stiffness of the specimens under a controlled force stress/strain testing protocol. Under the present DMA testing protocol, the average relaxation modulus was 9.40 MPa for the undoped group and 5.21 MPa for the Ni-doped group, as shown in Table . The DMA-derived relaxation modulus was higher than the quasi-static tensile modulus for both groups, indicating that the apparent stiffness of sodium-alginate-based artificial muscle strips depends on the testing protocol. The quasi-static tensile data were used for finite-element calibration, while the DMA results provide a complementary protocol-dependent mechanical comparison.

5. Supplementary DMA-Derived Relaxation Modulus under Controlled-Force Stress/Strain Testing.

Group Specimen Relaxation modulus/MPa Mean/MPa
Undoped SA-1 10.47285714 9.39994733
  SA-2 7.926818182  
  SA-3 9.800166667  
Ni-doped Ni-1 5.708833333 5.214869444
  Ni-2 3.6904  
  Ni-3 6.245375  
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3.3. Finite Element Calibration Based on Small-Strain Tensile Response

A finite element model was established to calibrate the small-strain tensile response of the sodium alginate-based artificial muscle strips, as shown in Figure . The calibration was restricted to the 0–5% engineering strain range, where the experimental stress–strain curves were approximately linear.

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Finite element model of the layered sodium alginate-based biomimetic artificial muscle strip: (a) layered geometry, material domains, and specimen dimensions; (b) boundary conditions and prescribed tensile displacement under the plane-stress assumption; (c) local mesh distributions at the fixed and loaded ends, showing the graded mapped quadrilateral mesh. The middle section in (c) is omitted for clarity.

3.3.1. Calibration of Stress–Strain Response Within 0–5% Strain

The finite element stress–strain curves were obtained from the reaction force at the loaded end and the applied displacement (Figure ). The engineering stress was calculated by using the initial cross-sectional area, and the engineering strain was calculated from the applied displacement divided by the gauge length. This treatment is consistent with the experimental data-processing method.

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Finite element calibration against the quasi-static tensile responses within the 0–5% engineering strain range: (a) undoped specimens (SA-1, SA-2, and SA-3); (b) Ni-doped specimens (Ni-1, Ni-2, and Ni-3). The green curves represent the group-level finite element calibration results.

For the undoped group, the calibrated finite element model reproduced the group-level initial stiffness within the 0–5% strain range. For the Ni-doped group, the model captured the average small-strain tensile response. The calibrated curves represent the group-level responses used for the mechanical parameter identification.

3.3.2. Mesh Sensitivity Analysis

Table summarizes the revised mesh convergence results. The global engineering stress changed from 0.048049 MPa for the coarse mesh to 0.048035 MPa for the fine mesh. The local maximum von Mises stress along the x = 0.5 mm cut line changed from 0.050353 to 0.050326 MPa, and the corresponding maximum axial strain changed from 0.052229 to 0.052266. Relative to the fine mesh, the medium mesh differed by 0.0139% in the local von Mises stress and 0.0057% in the local axial strain. Both values were far below the 2% convergence criterion. The medium mesh was used for subsequent finite element calculations.

6. Revised Mesh Sensitivity Analysis Based on Global and Local Mechanical Quantities.
Mesh Elements σFE at 5%/MPa Local σvm,max/MPa Local ε x ,max Δσvm/% Δεx/%
Coarse 1200 0.048049 0.050353 0.052229 0.0537 0.0708
Medium 4800 0.048039 0.050333 0.052263 0.0139 0.0057
Fine 19200 0.048035 0.050326 0.052266
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3.3.3. Deformation Characteristics and Stress Distribution

The deformation and von Mises stress distributions were analyzed at a 5% engineering strain. Both the undoped and Ni-doped models were dominated by axial elongation with no obvious bending or local instability in the gauge region.

At 5% engineering strain, both models remained dominated by the axial elongation. Most of the gauge region showed a comparatively uniform stress field, while higher stresses occurred near the fixed end and at interlayer transitions (Figure ). The maximum von Mises stress was 0.18936 MPa for the undoped model and 0.24912 MPa for the Ni-doped model. The latter value was approximately 31.6% higher than that under the same imposed displacement. The through-thickness profiles at x = 0.5 mm provide a more local comparison. Stress levels were relatively stable across the broad base film region, while sharper changes appeared across the thin interlayer regions. The Ni-doped calibration produced higher local peaks at these regions. The location of the stress concentration is controlled mainly by the fixed boundary and layered geometry; the calibrated material parameters modify the local stress magnitude.

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von Mises stress distribution of the layered sodium alginate-based biomimetic artificial muscle strips at 5% engineering strain: (a) undoped model; (b) Ni-doped model; (c) through-thickness von Mises stress profiles evaluated at x = 0.5 mm from the fixed end. The same color scale (0–0.25 MPa) is used in (a) and (b). The dashed vertical lines in (c) indicate the interfaces between adjacent material layers. The middle portions of the specimens in (a) and (b) are omitted for clarity. CL = Cross-linked gel.

3.4. Electro-Ionic-Mechanical Multiphysics Extension

The calibrated mechanical parameters were incorporated into a multiphysics model under a 3 V excitation. The model couples electrostatics, ion transport, concentration difference-driven differential swelling, and solid mechanics. The applied voltage establishes an electric potential field. Ion redistribution is described by the positive- and negative-ion concentrations c p and c n . Their difference, d c = c p – c n , represents the asymmetric ionic state associated with differential swelling. The resulting mechanical response was evaluated through the output force. A sequential solution strategy was used to improve the numerical stability and reduce the computational cost.

3.4.1. Coupling Framework and Representative Field Distributions

Figure summarizes the coupling pathways and representative field distributions. Under 3 V excitation, the electric potential spans the imposed voltage range across the active structure. The concentration-difference field at t = 60s shows regions with opposite signs, indicating an asymmetric ionic redistribution across the thickness. This ionic asymmetry provides the driving force for differential swelling in the mechanical model. The field plots demonstrate an internal coupling sequence.

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Multiphysics coupling framework and representative field distributions of the sodium alginate-based biomimetic artificial muscle under 3 V excitation: (a) coupling pathway from the applied voltage to the output-force response through electrostatics, ion transport, concentration difference, differential swelling, and solid mechanics; (b) electric potential distribution at t = 60s; (c) concentration-difference distribution, dc = c p – c n , at t = 60s.

3.4.2. Calibration Against the 3 V Output-Force Response

Figure compares the measured output-force response with the calibrated multiphysics model over 0–1000s. The model captures the initial rise, continued force development, and long-term approach to a plateau. The quantitative agreement is characterized by an RMSE of 0.0992 mN, an MAE of 0.0757 mN, R2 = 0.8152, and a plateau-force error of 3.76%. The results show that the mechanically calibrated framework reproduces the principal time-dependent output force features under 3 V excitation.

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Comparison between the experimental output-force response and the calibrated electro-ionic-mechanical multiphysics model under 3 V excitation over 0–1000s. The experimental measurements and their overall trend are compared with the calibrated multiphysics response. The agreement between experiment and simulation is quantitatively evaluated using the root-mean-square error (RMSE), mean absolute error (MAE), coefficient of determination (R2), and relative plateau-force error.

3.5. Scope of the Present Study

The statistical comparison was based on three independent tensile specimens per group and one previously optimized Ni-doped formulation. This small sample size limits the precision of the reported group mean moduli and the estimated between-group difference. This limitation is particularly relevant to the Ni-doped group, which showed substantial specimen-to-specimen variability (SD/mean = 42.4%) driven largely by the lower Ni-3 value. The nonsignificant result of Welch’s t test (p = 0.67) should be interpreted as inconclusive under the current sample size and not as evidence that the undoped and Ni-doped groups are mechanically equivalent. The conclusions are limited to the tested specimens and formulation. DMA provided supplementary protocol-dependent characterization, the mechanical calibration applies to the 0–5% engineering-strain range, and the electro-ionic-mechanical model was calibrated only under 3 V excitation.

4. Conclusion

This study combined quasi-static tensile testing, supplementary DMA characterization, small-strain finite element calibration, and an electro-ionic-mechanical multiphysics extension for sodium alginate-based biomimetic artificial muscle strips. The main conclusions are

  • (i)

    Both undoped and Ni-doped specimens showed approximately linear stress–strain responses within 0–5% engineering strain. This interval is suitable for an equivalent small-strain modulus extraction and linear elastic calibration.

  • (ii)

    The equivalent moduli were 0.975 ± 0.212 MPa for the undoped group and 1.118 ± 0.474 MPa for the Ni-doped group. No statistically significant between-group difference was detected (Welch’s t test, p ≈ 0.67); with n = 3 per group and substantial variability in the Ni-doped group, this result is inconclusive and should not be interpreted as evidence of equivalence.

  • (iii)

    The DMA-derived relaxation moduli differed from the quasi-static tensile moduli, confirming that the apparent stiffness is dependent on the testing protocol. The DMA results provide supplementary protocol-dependent mechanical characterizations.

  • (iv)

    The plane-stress finite-element model reproduced the group-level initial tensile stiffness. The mesh study showed the convergence of both global and local quantities, and the 4800-element mapped quadrilateral mesh was sufficient for subsequent calculations. Stress concentrations were controlled mainly by the fixed boundary and interlayer geometry, while calibrated material parameters affected their magnitudes.

  • (v)

    The calibrated mechanical framework was extended to a 3 V electro-ionic-mechanical model. The model reproduced the main 0–1000s output-force evolution with RMSE = 0.0992 mN, MAE = 0.0757 mN, R2 = 0.8152, and a plateau-force error of 3.76%. The model provides a basis for subsequent voltage-dependent studies.

Supplementary Material

ao6c03650_si_001.pdf (185.7KB, pdf)

Acknowledgments

This work was supported by The Natural Science Foundation of Hei-longjiang Province of China (Grant No. LH2020E065). We would like to thank our families for their constant support and encouragement throughout this research.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.6c03650.

  • Geometric and effective material parameters, boundary conditions, and mesh configurations of the mechanical finite element model; electrostatic, ion-transport, and time-dependent actuation parameters for the 3 V multiphysics model (PDF)

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

The Natural Science Foundation of Hei-longjiang Province of China under Grant No. LH2020E065.

The authors declare no competing financial interest.

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