*2.2. Cantilever Beam Theory*

Since the *JKR* theory could only describe the micro-scale adhesion properties and cannot characterize the elastic deformation of the setae rods, it may lead to the overestimation of the adhesion strength. When analyzing the microscopic contact of setae, a single seta rod could be regarded as a large flexible beam, and the end could be simplified as a viscous elastic sphere. This simplified model could deal with issues related to setae contact.

By applying vertical pressure on the setae, they can gradually achieve close contact with the surface. The setae would then be in the adsorption state. After that, a push-off force could be applied on the setae to separate them from the contact surface. Based on Figure 1, the mechanical analysis of a single polyurethane seta was carried out.

**Figure 1.** Schematic diagram of the stiffness of a single seta.

Neglecting the instability, when the setae were compressed in their axial direction, the ends of the setae were also subjected to a reaction force in the axial direction. According to the deformation theory of large flexible beams in the mechanics of materials, the deformation of the seta in their axial direction was:

$$
\Delta l = \frac{Fl}{\pi r^2 E\_1} \tag{16}
$$

where *l* is the length of the setae rod, *E*1 is the elastic modulus of the setae material, *r* is the diameter of the setae rod, and *F* is the axial load on the setae. The above Equation (16) could be transformed into a function *F* = *Ka*Δ*l*, where *Ka* is the stiffness of the setae along the axial direction, with the expression:

$$K\_a = \frac{\pi r^2 E\_1}{l} \tag{17}$$

when the end of the setae is subjected to an external force along its radial direction, the bending deformation of the material would cause a relative displacement Δ*h* in the radial direction. Therefore:

$$
\Delta h = \frac{Fl^3}{3E\_1 I} \tag{18}
$$

The above formula could be transformed into a function *F* = *Kp*<sup>Δ</sup>*h*, where *Kp* is the stiffness of the setae along the radial direction, and its expression is:

$$K\_p = \frac{F}{\Delta h} = \frac{3\pi r^4 E}{4l^3} \tag{19}$$

The longitudinal stiffness *Ky* of the setae at the inclination angle *θ* could be expressed as:

$$K\_{\rm y} = \frac{F\_{\rm y}}{\delta\_{\rm y}} = \frac{K\_P \cdot K\_a}{K\_P \sin^2 \theta + K\_a \cos^2 \theta} \tag{20}$$

In the same way, the expression of the lateral stiffness *KX* of the setae could be expressed as:

$$K\_{\mathbf{x}} = \frac{F\_{\mathbf{x}}}{\delta\_{\mathbf{x}}} = \frac{K\_P \cdot K\_{\mathbf{a}}}{K\_P \cos^2 \theta + K\_{\mathbf{a}} \sin^2 \theta} \tag{21}$$

To facilitate the study of the plastic deformation of the seta rod, the axial stiffness and the radial stiffness of the inclined seta rod were converted into the longitudinal stiffness *Ky* and the transverse stiffness *KX*. The comprehensive deformation of the seta rod could be decomposed into the *X*-direction deformation and the *Y*-direction deformation according to the coordinate system.

#### *2.3. Quasi-Static Contact Theory*

When the setae were pressed in or pulled out vertically at a low speed, the contact state of a single polyurethane seta with the surface could be approximately regarded as the quasistatic contact. As shown in Figure 2, the distance between the base of the setae and the contact surface was *Z*, the length of the setae rod was *L*, and the radius of the end ball was *R*. The seta was displaced as *y* in the *Y* direction. The initial distance of the setae from the contact plane was *h*0. The distance between the end of the seta and the contact surface during the movement was *h*.

$$h = Z - L\sin\theta - R - y\tag{22}$$

**Figure 2.** Schematic diagram of single polyurethane setae and surface.

The setae contact state was judged by analyzing the size of *h*. Since the interaction force of the setae in the process of contact and desorption was different, it was necessary to analyze the mechanical state of the pressing and desorption of the polyurethane setae.

#### 2.3.1. Quasistatic Indentation Force Analysis of Setae

The polyurethane setae were pressed vertically in the *Y* direction. Due to the van der Waals forces acting at a small distance, the polyurethane setae were just in contact with the surface without deformation at that time *h* = 0. The interaction force between the polyurethane setae and the surface was 0 at this time. When *h* < 0, the polyurethane setae were squeezed, and the interaction force between them was expressed as a repulsive force *F* > 0.

The displacement of the base of the setae was *y*, and the contact depth between the ball at the end of the setae was *y*1. When the rod of the setae was compressed, its elastic deformation was *y*2. As shown in Figure 3a, the relationship between their displacements could be obtained:

$$
\tau = y\_1 + y\_2 \tag{23}
$$

*y*

**Figure 3.** Static relationship between seta and contact surface;( D **a**) variation of a single seta press-in displacement;(**b**) force analysis of a single seta in contact with the surface.

As the setae moved down, the displacement *y*1 for the end ball was the *JKR* theoretical depth of contact:

$$y\_1 = \delta\_{JKR} \tag{24}$$

Based on the force balance theory, as shown in Figure 3b, we could obtain:

$$F = P\chi\kappa = y\_2 \times K\_y \tag{25}$$

By combining Equations. (7), (10), (20), (23), and (25), the relationship between the contact force and the displacement of the setae in the *Y* direction could be obtained.

During the pressing process, the seta would produce a lateral deformation *x*, which could be approximated according to the deformation coordination relationship:

$$\mathbf{x} = (y - y\_1) \times \tan \theta \tag{26}$$

The coefficient of friction between the setae and the surface was *μ*, and the setae experienced a leftward frictional force *Fμ* during the press-in process:

$$F\_{\mu} = \mu \times F \tag{27}$$

If the force generated by the lateral deformation of the setae was less than the friction force, that is *xKx* ≤ *μF*, then the setae would not produce a lateral displacement. Additionally, the friction force value was equal to the static friction. If the force generated by the lateral deformation of the setae was greater than the friction force, that is *xKx* > *μF*, then the setae would produce a lateral displacement. Additionally, the friction force value at this time was equal to the size of the kinetic friction force.

#### 2.3.2. Quasistatic Push-off Force Analysis of Setae

The polyurethane setae were perpendicular to the surface in the *Y* direction. When *h* < 0, the polyurethane setae were squeezed, and the interaction force between them was expressed as the repulsive force *F* > 0. In the *Y* direction, the contact force between the setae and the surface was the same as that in the pressed state. The friction force was opposite to that in the pressed state in the *X* direction.

When *δs* ≥ *h* > 0, the polyurethane setae end beads adhered to the surface, and the interaction force between them was an attractive force *F* < 0, as shown in Figure 4. Equation (25) was also applicable to the solution of the contact force in the desorption state, but the force state was opposite to that in the press-in state.

**Figure 4.** Schematic diagram of desorption of single polyurethane setae.

When *h* > *δs* the polyurethane setae were separated from the surface, and the interaction force between them was *F* = 0.

#### **3. Results and Discussion**

#### *3.1. JKR Model Analysis*

By solving the normalized *JKR* model of the setae contact, the relationship between the contact circle radius, contact force, and contact depth was obtained. Through the analysis of the curve characteristics, the dynamic mechanical relationship between the two objects in the mutual adsorption and desorption process was obtained.

The relationship between the force and the radius when the setae were in contact with the surface is shown in Figure 5a. It could be seen from the figure that the maximum adsorption force was at point C. The corresponding adsorption force was −1.5 μN at this point, and the radius of the contact circle was 1.24 μm. This point was the critical point where the ball end of the setae and the contact surface were separated from each other.

When the ball at the end of the setae was 0 from the surface, the adsorption force changed from point O (0, 0) to point A (0, −1.35 μN) in Figure 5b. The increase in the contact depth led to a growth in the adsorption force, which changed in the AB direction from point A. When the ball surface and the contact surface were separated from each other, the force would gradually decrease along with BA. The positive pressure gradually became the adsorption force, and then changed from point A to point D (−0.8, −0.75). When it was further detached from point D, the adsorption force would change suddenly from point D (−0.8, −0.75) to point E (−0.8, 0), which also meant that the polyurethane setae were completely detached from the contact surface. Point D was the critical point, where the polyurethane setae and the contact surface were separated from each other.

**Figure 5.** The dynamic mechanical relationship between the sphere and the contact surface: (**a**) the relationship between the radius and the force; (**b**) the relationship between the contact force and the contact depth.

#### 3.1.1. Influence by Elastic Modulus

Based on the above theory, the effect of the material's elastic modulus and surface energy on the interaction force model was further analyzed. The dynamic mechanical simulation analysis was carried out by selecting real polyurethane seta material parameters. The sphere and the plane were selected to be in contact with each other, the radius of the sphere at the end of the polyurethane setae was 4 μm, the elastic modulus was 1.413 Mpa, the Poisson's ratio was 0.3 mJ/m2, and the adhesion energy was 40 mJ/m2. The elastic modulus of the contact plane was 55 Gpa, the Poisson's ratio was 0.25, and the adhesion energy was 170 mJ/m2.

Taking the elastic modulus as 0.1 Mpa, 1 Mpa, and 10 Mpa for the dynamic simulation, we could obtain the relationship curve between the circle radius and the external load, and the external load and the depth. It could be seen from Figure 6a that with the increase in the elastic modulus, the contact circle radius under the same contact force decreased significantly from 3.69 μm at 0.1 Mpa to 0.74 μm at 10 Mpa. An increase in the elastic modulus resulted in less deformation under the same contact force. Under the same contact force, the deformation became smaller as the elastic modulus increased. It was also found that the maximum push-off force of spheres with different elastic moduli experienced little change during the separation process.

**Figure 6.** Dynamic analysis results under different elastic moduli of 0.1 Mpa, 1 Mpa, and 10 Mpa: (**a**) the relationship between the contact radius and the contact force under different elastic moduli; (**b**) the relationship between the contact force and the contact depth under different elastic moduli.

#### 3.1.2. Influence by Material Surface Energy

Taking the sphere adhesion energy as 4 mJ/m2, 40 mJ/m2, and 100 mJ/m2, the dynamic simulation was carried out. The relationship between the radius of the circle and the load, and the load and the depth were obtained. It could be seen from Figure 7a that with the increase in the surface adhesion energy, the maximum adsorption force when the setae were desorbed increased significantly. The maximum adsorption force increased from 0.71 μN at 4 mJ/m<sup>2</sup> to 3.55 μN at 100 mJ/m2. With the increase in the surface adhesion energy of the contacting object, the corresponding contact depth also increased gradually during desorption, from 0.23 μm at 4 mJ/m<sup>2</sup> to 0.71 μm at 100 mJ/m2.

**Figure 7.** Dynamic analysis results under the conditions of different surface energies of 4 mJ/m2, 40 mJ/m2, and 100 mJ/m2: (**a**) the relationship between the contact radius and force under different surface energies; (**b**) the relationship between the contact force and depth under different surface energies.

To sum up, the contact depth would increase with the decrease in the elastic modulus under the same contact load, but had little effect on the maximum push-off force. The depth of the contact and maximum push-off force would increase as the surface energy of the contacting object increased. Therefore, the main factor affecting the setae push-off force was the surface energy of the material. When the material for creating the setae was determined, no matter how the morphology and size of the setae changed, the value of the push-off force was constant.

#### *3.2. Simulation of the Single Seta*

The interaction force between a single seta and the surface adhesion or detachment had different forms, so the micro-adhesion state of the setae needed to be considered and analyzed separately. When *δ* > 0, the seta would be in contact with the surface and elastically deform. When *δ* < 0, the polyurethane seta would undergo a plastic deformation due to the effect of adhesion and remained in contact with the surface. Until the elastic force overcame the adhesion *δ* > *δ*0, the setae detached from the surface.

The dynamic desorption process of a single polyurethane seta was simulated and analyzed, and the parameters were selected as shown in Table 1. The dynamic desorption process of setae was solved by Matrix Laboratory (MATLAB) programming.

**Table 1.** Selection of polyurethane setae parameters.


The relationship curve between the contact force and the contact depth was obtained through a simulation, as shown in Figure 8. The setae were slowly pulled out at a certain speed, and the contact force decreased with the contact depth and changes toward BA. When the setae were pulled apart to −7.2 μm, the contact was suddenly disconnected. From point D to point E, the adsorption force suddenly changed from 4.1 μN to 0 N. The EF segmen<sup>t</sup> setae were separated from the contact surface, and there was no interaction force detected.

**Figure 8.** Simulation curve of contact force and contact depth for setae array.

#### *3.3. Setae Array Simulation and Experiment*

#### 3.3.1. Rough Surface Model

An important assumption of the classical contact theory is that the contact surface is geometrically smooth, but the real surface in engineering is rough. Therefore, it was necessary to establish a roughness model that could approximate the actual surface and could be easily calculated. As shown in Figure 9, assuming that the base surface of the setae was parallel to the reference plane of the contact surface, the spacing was *Z*, the length of each seta was *L*, the setae were parallel to each other, and the angle between the setae and the base surface was θ.

**Figure 9.** Roughness model.

It was assumed that the peak height distribution of the contact surface conformed to a normal distribution, and the reference surface of the rough surface was a plane. The distance was *h* between the contact surface profile and the reference plane, and the height satisfied the following Gaussian distribution:

$$\mathbf{g}(h) = \frac{1}{\sqrt{2\pi\sigma}} e^{-\frac{h^2}{2\sigma^2}}\tag{28}$$

3.3.2. Dynamic Simulation and Experimental Verification

The dynamic adhesion process of polyurethane setae arrays was simulated and analyzed, and the setae were pulled out vertically from the bottom to the top. When the material area was 1 cm<sup>2</sup> and the seta spacing was 4 μm, there was 1 × 10<sup>7</sup> setae in total, and the inclination angle of the seta was 60◦. The elastic modulus was taken as 1.413 Mpa, the Poisson's ratio was 0.3, the surface adhesion energy was 40 mJ/m2, and the friction coefficient between the end of the setae and the surfaces was 0.2. The elastic modulus of the contact surface was taken as 55 Gpa, the Poisson's ratio was 0.25, and the adhesion energy was taken as 170 mJ/m2. The mean square error of the surface roughness was 1. Using MATLAB programming to solve the dynamic desorption process of setae, the results obtained by dynamic calculation are shown in the following Figure 10.

**Figure 10.** Simulation curve of the contact force with different contact depths.

The relationship between the push-off force and the depth was obtained through a simulation, as shown in Figure 10. With the continuous pulling out of the setae array, the contact force decreased gradually from 8.6 N. When the contact depth was less than 0 μm, the direction of the contact force changed from pressure to adsorption. The curve had an extreme point as the adsorption force reached a maximum value of −2.85 N. As the setae

pulled further apart, the adsorption force gradually decreased to 0 N, and all the setae were separated from the surface.

Figure 11a shows the relationship between the adsorption force and the preload on the setae array when the mean square error of the roughness was 1. We could see from the simulation curve that the adsorption force of the setae array increased rapidly with the increase in pressure. After the pre-pressure reached 1 N, the adsorption force stabilized at about 2.3 N. The experimental results matched well and the deviation error of the data was 7.81%.

As the preload force increased, the number of setae arrays in contact with the surface increased, so the preload force determined the distribution of adhesion between the array and the surface. Different adhesion distributions would produce different interaction force relationship curves during the detachment process. There was an extreme point of adhesion force in the desorption curve under a certain pre-pressure, as shown in Figure 10. By collecting the maximum desorption force under different pre-pressures, the relationship curve between the pre-pressure and the maximum desorption force was obtained. During dynamic detachment, the maximum adhesion force produced by the setae array increased with an increasing preload, as shown in Figure 11a. The preload was further increased, and the maximum adhesion attraction tended to reach a stable value. Because the area of the material was constant in both experiments and simulations, there was a saturation value for the number of setae contacts. The mean square error of surface roughness was selected as 0.5, 1, and 1.5, respectively. The experimental and simulation results of setae in contact with surfaces of different roughness are shown in Figure 11b. The root mean square (RMS) was used to describe the dispersion between the experimental data and the simulation data, and the calculation result was 15.9% [21]. The peak adsorption force increased continuously at different preloads as the root mean square deviation of the surface decreased. The contact of the polyurethane setae array with the rough surface became better as the root mean square deviation decreased, so the smooth surface was more favorable to the adsorption of the setae array.

**Figure 11.** The relationship curves of the adsorption force between experiment and simulation: (**a**) results data with different preloads; (**b**) results data with different surface roughness.
