**1. Introduction**

In recent years, substantial progress has been achieved in gecko-inspired adhesive technology since the discovery of the uniquely layered footpad structure of geckos [1,2]. Geckos can climb on almost any surface, or even stand upright or upside down, because their layered structure is composed of inclined villi, and the end of each villus is composed of many thin setae [3]. Researchers have been able to understand the adhesion mechanism of gecko feet and have promoted the development of biomimetic adhesion materials [4–6]. Based on the van der Waals force contact principle, biomimetic adhesive materials may exhibit stable contact performance in complex environments [7,8]. The van der Waals force exists in the molecules on the surface of various objects without any requirements regarding the environment, so it is employed by animals such as geckos to adhere to the surfaces of various objects [9].

A lot of research has been carried out on the design and production of gecko-like adhesive materials and structures at home and abroad. In 2000, Autumn et al. [10,11] studied the adhesion mechanism of gecko setae. In the United States, there has been an upsurge in research on biomimetic dry adhesive materials and structures. The adhesion mechanism of gecko setae dominated by the van der Waals force and supplemented by

**Citation:** Lin, Q.; Wu, C.; Yue, S.; Jiang, Z.; Du, Z.; Li, M. Dynamic Simulation and Parameter Analysis of Contact Mechanics for Mimicking Geckos' Foot Setae Array. *Crystals* **2022**, *12*, 282. https://doi.org/ 10.3390/cryst12020282

Academic Editors: Yong He, Wenhui Tang, Shuhai Zhang, Yuanfeng Zheng, Chuanting Wang, George Z. Voyiadjis and ShujunZhang

Received: 23 January 2022 Accepted: 16 February 2022 Published: 18 February 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the capillary force has been widely accepted. However, the elastic deformation of the setae was rarely involved, leading to the overestimation of the adhesion to the setae. In 2003, Geim et al. proposed a "gecko tape" material, which is fabricated with an array of geometrically shaped flexible plastic posts. The structure has a repeatable attachment function and self-cleaning capability [12]. Persson proposed a simple model to study the effect of surface roughness on the adhesion of gecko foot setae. This model simplifies the tongue depressor at the end of the setae and regards the tongue depressor as an equivalent plane. Assuming that the surface roughness follows a normal distribution, the simulation results show whether the adhesion depends on the magnitude of the surface roughness [13,14]. The literature carries out experimental and theoretical analyses on adhesion to rough surfaces, but does not emphasize the change in the adhesion force of microscopic setae during adhesion. In 2005, the Max Planck Institute for Metal Research in Germany simulated setae found on the foot of a gecko, then optimized and simulated the pulling process of the gecko setae, but lacked research on the adhesion properties of array materials [15,16]. Dai Liming et al., of the University of Dayton in the United States, recently used chemical vapor deposition to obtain high-density, large aspect ratio carbon nanotube-adhered arrays [17]. The literature did not further elaborate on the adhesion principle of the array material. Guo et al. studied the dry adhesion of VACNTs in different temperature ranges and explained the phenomenon of adhesion with temperature [18]. The microscopic morphology of the material was optimized by Mark Cutkosky et al. at Stanford University to develop a millimeter-scale array with sharp ends. The adhesion strength of the experimentally obtained adhesion array is about 0.24 N/cm2, with obvious anisotropy, but the connection between the microscopic setae adhesion theory and the macroscopic adhesion has not been explained further [19,20].

Research on the adhesion characteristics of geckos has good references regarding the development of bionic adhesion materials. Therefore, we could explore the principle of the microstructure of gecko feet to study the adhesion characteristic of polyurethane setae array materials. The traditional Hertz contact theory is mainly used for the contact between two elastomers. When the research scale is further reduced and the surface energy is introduced, the Hertz contact theory cannot be explained. Based on the Hertz theory, the Johnson–Kendall–Robert (*JKR*) contact theoretical model is established by introducing surface adhesion. The dynamic mechanical properties of setae in microcontact are further studied by integrating the *JKR* contact theory model with the cantilever beam model. Through the established mathematical model, the influence mechanism of roughness, surface energy, and elastic modulus on the microcontact dynamic characteristics of polyurethane, the setae array is analyzed.

The structure of this paper is as follows: Section 1 summarizes the relevant research and the main contributions of this work. In Section 2, the microscale adhesion theory and cantilever beam theory are studied and integrated to form the setae quasistatic contact theory. Descriptions of experiments, simulations, and the model analysis are included in Section 3. Section 4 concludes the whole research.

#### **2. Micro Contact Theory of Setae**

#### *2.1. Microscale Adhesive Contact Theory*

In the macroscopic theory, when objects are in contact with each other, the elastic force is much larger than the surface force of the objects. Thus, the surface force is often ignored. However, when the characteristic scale of the research object is reduced to a certain range, there can be many phenomena that cannot be explained by the traditional macroscopic contact theory. The reason is that the role of the surface force and surface energy between two objects is the key factor in determining the adhesion, contact, and deformation behavior of solid surfaces. In areas where surface forces are dominated, traditional continuum mechanics methods are no longer applicable. The concept of surface force was introduced in classical mechanics; thus, forming the theory of the microscale adhesive contact.

In the Hertz contact model, due to the lack of an adhesion force, when the applied external load is *Ph*, a contact circular surface with a radius *ah* would be generated, and its contact depth *δh* and pressure distribution *Ph*(*r*) in the contact area could be given by the following formula:

$$a\_h^3 = \frac{3RP\_h}{4E^\*}\tag{1}$$

$$a\_h^3 = \frac{3RP\_h}{4E^\*}\tag{2}$$

$$p\_h(r) = \frac{3P\_h}{2\pi a\_h^2} (1 - \frac{r^2}{a\_h^2})^{1/2} \tag{3}$$

where *E*∗ is the equivalent elastic modulus, *E*∗ = ( <sup>1</sup>−*<sup>ν</sup>*21 *E*1 + <sup>1</sup>−*<sup>ν</sup>*22 *E*2 )−1 and *R* are the equivalent radius 1*R* = 1*R*1 + 1*R*2 , *R*1*R*2 are the radii of two contact spheres, respectively, and *r* is the distance between the contact point and the center of the contact area.

Based on the Hertz contact model, the adsorption force *Pa* between the contact surfaces is introduced, and its pressure distribution *Ph*(*r*) in the contact area could be expressed by the following formula:

$$P\_a(r) = \frac{P\_a}{2\pi a^2} (1 - \frac{r^2}{a\_h^2})^{-1/2} \tag{4}$$

The adsorption term *Pa* has a crack propagation singularity at the contact boundary, and the corresponding pressure increase coefficient is *KI* = *Pa* <sup>2</sup>*a*√*πa* . Using the Irwin formula, the relationship between the adsorption energy and the adsorption energy could be obtained as:

$$G = \frac{K\_I^2}{2E^\*} = w\tag{5}$$

Among them, *G* is the elastic energy release rate, *w* is the adhesion energy, and *w* = <sup>2</sup>√*<sup>γ</sup>*1*γ*2, *γ*1, and *γ*2 are the surface free energy of the two objects in contact with each other, respectively. The expression of the adsorption term could be obtained from the above formula:

$$P\_4 = \sqrt{8\pi w E^\* a^3} \tag{6}$$

The final *JKR* theoretical contact pressure distribution can be obtained:

$$P\_{IKR} = P\_h - P\_a = \frac{4E^\*a^3}{3R} - \sqrt{8\pi w E^\*a^3} \tag{7}$$

The expression of the contact circle radius *aJKR* is:

$$
\sigma\_{JKR}^3 = \frac{3R}{4E^\*} (P + 3\pi wR + \sqrt{6\pi wRP + \left(3\pi wR\right)^2})\tag{8}
$$

The corresponding normal displacement is *<sup>δ</sup>JKR* and could be written as:

$$
\delta\_{JKR} = \delta\_{\rm li} + \delta\_{\rm d} \tag{9}
$$

where *δh* is the Hertz compression depth, *δa* is the adsorption compression depth, and *δa* = *Pa* 2*πE*<sup>∗</sup> = √<sup>8</sup>*πE*<sup>∗</sup>*wa*<sup>3</sup> 2*πE*<sup>∗</sup> ; finally, the normal displacement could be written as:

$$\delta\_{fKR} = \frac{a\_{fKR}^2}{R} - \sqrt{\frac{2\pi \pi w a\_{fKR}}{E^\*}} \tag{10}$$

To facilitate the analysis of the essence of the *JKR* contact model, the obtained dimensionless external load *P*<sup>∗</sup>, contact circle radius *<sup>a</sup>*<sup>∗</sup>, and contact depth *δ*∗ are:

$$P^\* = \frac{P\_{IKR}}{\pi R w} \tag{11}$$

$$a^\* = a\_{J\&R} \left(\frac{4E^\*}{3\pi R^2 w}\right)^{1/3} \tag{12}$$

$$
\delta^\* = \delta\_{\rm JKR} \frac{9\pi^2 R w^2}{16E^{\*2}} \tag{13}
$$

Then, according to the formula (7), the expression of dimensionless load *P*∗ could be obtained as:

$$P^\* = P\_h^\* - \left(6P\_h^\*\right)^{1/2} = a^{\*3} - \left(6a^{\*3}\right)^{1/2} \tag{14}$$

Based on Equation (10), so that the contact depth *δ*<sup>∗</sup>:

$$
\delta^\* = a^{\*2} - 2\left(\stackrel{2a^\*}{\quad} \stackrel{\cdot}{\quad} \stackrel{\cdot}{3}\right)^{1/2} \tag{15}
$$

When the external load is 0, the contact radius generated by the van der Waals force is *a*0 = (<sup>6</sup>*πR*2*<sup>w</sup>* \$ *K*)1/3, and the following could be seen from Equation (8): the maximum adhesion force: *Fad* = −1.5*πRw*; the separation radius: *as* = 0.63*a*0; the normal displacementduringseparation:*δs*=−0.21*<sup>a</sup>*02*R*−1.
