HYPERELASTIC SIMULATION FOR ACCURATE PREDICTION OF GECKO INSPIRED DRY ADHESIVE DEFORMED SHAPE AND STRESS DISTRIBUTION PRIOR TO DETACHMENT B. Bschaden, B. Ferguson, and D. Sameoto University of Alberta Edmonton, Alberta, T6G 2G8, Canada [email protected]
Introduction Biomimetic dry adhesives have been of interest to researchers for the last decade with many applications which could benefit from the reversible adhesion to surfaces such materials provide. It is known that the shape of gecko inspired dry adhesives fibers greatly affects their performance with mushroom shaped fibers generally providing the highest adhesion strengths . Accurately designing optimal shapes however has been a challenge, partially limited by manufacturing technologies and an incomplete understanding of mechanical behavior. Previously published models mainly examine the stresses and physics of the fiber cap as it adheres to a surface. An analytical model based on the crack propagation energy barrier  provides insight to the correlation between adhesion strength and cap overhang. This work was extended to use finite element analysis (FEA) to determine optimized fiber shapes based on the adhesion behavior model developed . Kim et al.  also studied the normal stress distribution of a mushroom shaped fiber, further demonstrating the importance of overhanging caps to enhance adhesion. While Tang et al.  included the effects of viscoelasticity, to the authors’ knowledge, all previous models assume linear elastic material behavior. We show that strongly adhering bio-inspired adhesives can demonstrate significant elongation (~150%) of loaded fibers, making previously published linear elastic simulations unrealistic for elastomer based synthetic dry adhesives. To better capture the hyperelastic behavior of rubber materials, strain energy functions  are often used. This work uses the Mooney-Rivlin strain energy equation to model the material in a finite element package. The refined material model allows a more accurate examination of the stress distribution throughout the fiber during loading, including the effect of geometric changes.
microscope (SEM). The fiber geometry used for the 3D model was accurately estimated from these SEM images. The composite consists of a similar (anti-static) version of ST-1060 which slows the curing process of the polymer, making the addition of carbon black (Vulcan XC-72) easier. The anti-static ST-1060 also has a lower durometer, making the filled composite more closely resemble that of standard ST-1060. Using a custom built test system  the adhesive strength of the fibers was determined. Multiple trials were completed to attain the average adhesive pressure prior to pull off, which along with an estimate of the average number of fibers in contact, was converted to an adhesive force per fiber. The experimentally determined adhesion strength of ~0.58 mN for each 24m diameter fiber, is utilized for the semi-empirical FEA model developed in this work. Material Property Testing To determine the hyperelastic behavior of the polymer, a uniaxial test, ASTM standard D412 was performed using an Instron testing machine, model number 5943 with 2713-011 self-tightening clamps. A pure uniaxial test was sufficient for the model developed because the fibers are predominantly loaded axially. Using the Mooney-Rivlin strain energy function, described in detail in , the two model constants C1 and C2 were determined numerically from the experimental data recorded. Figure 1 shows an example of experimental stress-strain data and the data fit Mooney-Rivlin defined stress-strain curve for ST-1060 as well as experimental data for Sylgard 184.
Fabrication and Testing Dry Adhesive Fibers The dry adhesives modeled in this study were fabricated as previously described in  but with uncollimated 254 nm exposure for patterning the acrylic master mold. Adhesives were fabricated from ST-1060 polyurethane as well as anti-static ST-1060/carbon black composite to allow for viewing in scanning electron
Figure 1. Mooney-Rivlin Strain energy fit vs. experimental data for ST-1060 Polyurethane Elastomer and experimental data for Sylgard 184.
Table 1 gives a summary of the Young’s Modulus and the Mooney-Rivlin constants for the tested materials and composites. Note the Young’s Modulus was determined during 1-2 mm extension of a 36 mm test length sample. Table 1. Instron tensile testing results for various polymers and polymer composites. Material ST-1060 ST-1060 Anti-Static ST-1060 Anti-Static/4%CB ST-1060 Anti-Static/8%CB ST-1060 Anti-Static/12%CB Sylgard 184 Sylgard 184/8% CB
Modulus (MPa) 3.1 3.7 4.0 5.0 6.1 2.8 6.4
C1(kPa) 330 200 450 530 650 NA NA
C2 (kPa) 360 400 320 240 570 NA NA
Values for C1 and C2 for Sylgard are not given as some were found to be negative. As shown by Johnson , negative Mooney-Rivlin constants can produce unstable material models and should not be used. This effect may be due to an artifact of the experimental setup with Sylgard being extremely sensitive to slight slipping of the sample within the clamp or mold imperfections.
Model Setup A full 3D model of the dry adhesive fibers was developed to account for the square shaped cross section of the pillar base. This non-circular profile results from reduced exposure dose during fabrication from uncollimated light through the smallest gaps between pillars in the square array. The fiber geometry was modeled with SolidWorks and imported for meshing and solving in COMSOL. Figure 2 a) illustrates the model created compared to the SEM image in b) of the fiber.
singularities at the perimeter of the cap while frictionless modeled surfaces can have slightly compressive loads at the cap perimeter. A roller boundary condition is physically inappropriate because the fibers are known to be able to resist shear loading, as shown in Figure 3. This fiber maintained adhesion in shear even in the presence of a defect in the fiber cap; suggesting that the center of the fiber cap provides most of the adhesion.
Figure 3. SEM image of a fiber showing shear adhesion with the presence of a defect in the interface. The SEM image in Figure 4 b) shows significant shrinkage of the cap (20.3 m vs. 24 m designed diameter) at large fiber extensions. This shrinkage also demonstrates that the “fixed” interface assumption is invalid. The appropriate boundary condition allows for restrained shrinkage of the cap while adhesion is maintained. The desired boundary condition between the fiber cap and the adhered surface can be estimated by a simple Coulomb friction model. We can estimate the frictional sliding resistance force, Ff similar to the method used by Zaghloul et al.  to determine the friction in an AFM tip. As described in equation 1, the net interfacial force is the difference between the Van der Waals adhesion force, FVdW, assumed to be ~1 MPa, and the normal reaction force at the element, Fnormal, due to applied loading, multiplied by the static coefficient of friction, which must be determined experimentally for specific surfaces. [
Figure 2. a) CAD model from SolidWorks with dimensions in m measured from b) SEM image of dry adhesive fiber. The boundary conditions were developed to best match the physical adhesives studied in this work. A distributed load is applied at the end of a 100 m thick backing layer, modeled with a square cross section equal to the pitch of the fibers. Applying the load at a distance from the fiber significantly alters force lines through the fiber. Both fixed and roller cap interfacial boundary conditions are examined. As shown previously in , a fixed surface boundary condition produces stress
While equation 1 describes the desired boundary condition, its application into a FEA package produces convergence issues. The authors are currently working on a refined model to incorporate this condition. FEA results given in Table 2 suggest that the actual condition is quite close to the frictionless roller situation making it a good approximation to determine fiber stress. To validate the model, a SEM image of a loaded fiber was compared to the geometry of the FEA model with the experimentally estimated load applied. Figure 4 shows that the model is a good match to the actual geometry of the fiber.
Table 2. Summary of normal loading FEA 12% composite results compared to SEM imaging of loaded fibers. Model ST-1060/CB FEA roller ST-1060/CB FEA fixed ST-1060/CB FEA (circular model) roller ST-1060 CB SEM
Figure 4. a) FEA of a loaded fiber and b) SEM of loaded fiber. Color scale represents normal stresses.
Results and Discussion To investigate stress distributions under normal loading conditions, a load of 0.58 mN was applied at the base of the 30 x 30 m backing material. Figure 5 shows the results for normal loading stress distribution for the fiber cross section and the cap interface. Note these results are for the ST-1060 12% carbon black composite material to allow for comparison with the fibers viewed in SEM. a)
Elongation (%) 151 147 147
5 1.9 2.1
(%) Cap shrinkage 6.3 0 6.3
Conclusions We have for the first time used empirical tests to determine the hyper-elastic behavior of ST-1060 polyurethane and composites and used these material properties in full 3D simulations of gecko-inspired adhesives. Our SEM and visual images reveal that, for strong adhesives, hyperelastic behavior occurs and requires that the deformed shape of the fibers before pull-off be investigated for understanding of real stress distribution. The Mooney-Rivlin hyperelastic model provides good agreement with our material properties and simulated fiber behaviors. Based on our measured performance and SEM images, we can state that while a frictionless roller provides a good estimate, the fiber caps cannot fully be modeled with fixed or slider conditions when experiencing large deformations, but rather a sliding surface that can maintain friction with a tensile force applied.
We would like to acknowledge Dr. Anastasia Elias for providing access to the Instron material test equipment, NSERC and Alberta Innovates for student funding, and Micralyne Inc. for supporting this work under the NSERC strategic grants program.
References Figure 5. 2D view of a), b) fixed cap interface and c), d) roller interface for fiber cross section and cap interface respectively. Color scale represents normal stresses. Ideally, the stress distribution on the top cap would be close to uniform in order to maximize adhesion in the ideal loading conditions. However, due to random defects throughout the surface from manufacturing or contamination after use, a higher stress concentration at the center of the fiber, as shown in Figure 5 d), may actually produce better performing micro-scale dry adhesives by leading to a higher incidence of cavitation type adhesion failures . As shown in Figure 5 b) and d), the stress distribution in the fiber cap is nearly circular, suggesting a 2D axisymmetric FEA model may be sufficient for studying the normal loading condition to save computational time. The results for the circular model along with a summary of other normal loading conditions are given in Table 2.
1. 2. 3. 4.
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