Cluster of the Kendall-type adhesive microcontacts as a simple model for load sharing in bioinspired fibrillar adhesives


The problem of multiple adhesive contact is considered for an elastic substrate modeled as a transversely isotropic elastic half-space. It is assumed that a large number of the Kendall-type microcontacts are formed between the substrate and circular rigid (i.e., nondeformable) and frictionless micropads, which are interconnected between themselves, thereby establishing a load sharing. The effect of microcontacts interaction is accounted for in the formulation of the detachment criterion for each individual microcontact. A number of different asymptotic models are presented for the case of dilute clusters of microcontacts with their accuracy tested against a special case of two-spot contact, for which an analytical solution is available. The pull-off force has been estimated and the effects of the array size and the microcontact spacing are studied. It is shown that the flexibility of the micropads fixation, which is similar to that observed in mushroom-shaped fibrils, significantly increases the pull-off force. The novelty of the presented approach is its ability to separate different effects in the multi-scale contact problem, which allows one to distinguish between different mathematical models developed for bioinspired fibrillar adhesives.

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The authors are grateful to the DFG (German Science Foundation—Deutsche Forschungsgemeinschaft) for financial support.

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Appendix A: Collins’ asymptotic solution of the two-spot contact problem

It was shown [25] that the contact pressure produced by the first punch whose center is located at the origin of coordinates can be represented in the form

$$\begin{aligned} p(r,\phi )=\pi E^{{*}}\sum \limits _{n=0}^\infty {\tau _n (r)\cos n\phi } , \end{aligned}$$

where \(\phi \) is the polar angle measured from the direction pointing to the center of the second punch,

$$\begin{aligned} \tau _n (r)=-\frac{r^{n-1}}{2\pi }\frac{\hbox {d}}{\hbox {d}r}\int \limits _r^a {\frac{s^{1-n}S_n (s)\hbox {d}s}{(s^{2}-r^{2})^{1/2}}} \end{aligned}$$

and \(S_n (s)\) is a polynomial with the first term of degree n. In particular, we have

$$\begin{aligned} S_1 (s)=-\frac{8\varepsilon \delta s}{\pi ^{2}d}\left\{ {1-\frac{2\varepsilon }{\pi }+\frac{1}{2}\left( {\frac{s^{2}}{d^{2}}+\varepsilon ^{2}+\frac{8\varepsilon ^{2}}{\pi ^{2}}} \right) } {-\frac{\varepsilon }{3\pi }\left( {\frac{3s^{2}}{d^{2}}+13\varepsilon ^{2}+\frac{24\varepsilon ^{2}}{\pi ^{2}}} \right) } \right\} . \end{aligned}$$

The contact force exerted by either punch on the half-space is given by Collins in the form of an asymptotic expansion

$$\begin{aligned} F_1 =F_2= & {} \pi E^{{*}}\int \limits _0^a {S_0 (s)\hbox {d}s} \nonumber \\= & {} 2E^{{*}}a\delta \left\{ {1-\frac{2\varepsilon }{\pi }+\frac{4\varepsilon ^{2}}{\pi ^{2}}} \right. \left. {-\frac{2\varepsilon ^{3}}{3\pi }\left( {1+\frac{12}{\pi ^{2}}} \right) +\frac{16\varepsilon ^{4}}{3\pi ^{2}}\left( {1+\frac{3}{\pi ^{2}}} \right) -\frac{4\varepsilon ^{5}}{\pi }\left( {\frac{1}{5}+\frac{14}{3\pi ^{2}}+\frac{8}{\pi ^{4}}} \right) +O(\varepsilon ^{6})} \right\} .\nonumber \\ \end{aligned}$$

The moment exerted by the first punch on the half-space can be evaluated as follows:

$$\begin{aligned} M_2^1= & {} -\int \limits _0^{2\pi } {\hbox {d}\phi } \int \limits _0^a {r\cos \phi p(r,\phi )r\hbox {d}r} \nonumber \\= & {} -\pi ^{2}E^{{*}}\int \limits _0^a {\tau _1 (r)r^{2}\hbox {d}r}. \end{aligned}$$

Here, \(\tau _1 (r)\) is given by Eqs. (A.2) and (A.3).

Substituting the expression for \(\tau _1 (r)\) given by formula (A.2) into the integral (A.5) and changing the order of integration, we obtain

$$\begin{aligned} M_2^1 =-\pi E^{{*}}\int \limits _0^a {sS_1 (s)\hbox {d}s}. \end{aligned}$$

Now, the substitution of (A.3) into (A.6) yields

$$\begin{aligned} M_2^1 =\frac{8}{3\pi }E^{{*}}a^{2}\varepsilon ^{2}\delta \left\{ {1-\frac{2\varepsilon }{\pi }+\frac{4\varepsilon ^{2}}{5\pi ^{2}}} ( {5+\pi ^{2}}) {-\frac{2\varepsilon ^{3}}{15\pi ^{3}}( {60+37\pi ^{2}} )+O(\varepsilon ^{4})} \right\} . \end{aligned}$$

Further, the calculation of terms in the series expansion (A.1) is reduced to the evaluation of the integral

$$\begin{aligned} J_m (r)=\int \limits _\rho ^a {\frac{s^{2m+1}\hbox {d}s}{(s^{2}-r^{2})^{1/2}}}. \end{aligned}$$

Using integration by parts, the following recurrent formula can be established:

$$\begin{aligned} J_{m+1} (r)=\frac{a^{2m+2}}{2m+3}( {a^{2}-r^{2}} )^{1/2}+\frac{(2m+2)}{2m+3}r^{2}J_m (r). \end{aligned}$$

At the same time, we have \(J_0 (r)=( {a^{2}-r^{2}})^{1/2}\).

The maximum (by absolute value) SIF of the contact pressure (A.1) is attained at \(\phi =\pi \) with the value

$$\begin{aligned} K_\mathrm{I} (\pi )=-\frac{E^{{*}}\delta }{\sqrt{\pi a}}k_\mathrm{I}^0 (\varepsilon ), \end{aligned}$$


$$\begin{aligned} k_\mathrm{I}^0 (\varepsilon )= & {} 1-\frac{2\varepsilon }{\pi }+\frac{4\varepsilon ^{2}}{\pi ^{2}}(1+\pi )-\frac{8\varepsilon ^{3}}{3\pi ^{3}}[ {3(1+\pi )+2\pi ^{2}} ]+\frac{4\varepsilon ^{4}}{3\pi ^{4}}[ {12(1+\pi )+\pi ^{2}(11+6\pi )}] \nonumber \\&-\,\frac{8\varepsilon ^{5}}{15\pi ^{5}}[ {60(1+\pi )+\pi ^{2}( {70+55\pi +22\pi ^{2}} )} ]+O(\varepsilon ^{6}). \end{aligned}$$

Appendix B: Proof of Theorem 1

We start with the equation

$$\begin{aligned} C\mathbf{F}=\delta \mathbf{e}, \end{aligned}$$

assuming that during loading the punches are rigidly connected and their flat bases maintain horizontal positions in one plane, so that the punches vertical displacement is denoted by \(\delta \).

From Eq. (B.1), it immediately follows that

$$\begin{aligned} \mathbf{F}=\delta C^{-1}{} \mathbf{e}, \end{aligned}$$

The detachment criterion [see Eqs. (9) and (76)]

$$\begin{aligned} {\mathop {\max }\limits _j} | {F_j } |=f^{{*}} \end{aligned}$$

can be rewritten in the form

$$\begin{aligned} \Vert \mathbf{F} \Vert _\infty =f^{{*}}, \end{aligned}$$

so that Eqs. (B.2) and (B.3) imply the following relation for the absolute value \(\delta _c \) of the critical displacement:

$$\begin{aligned} \delta _c =\frac{f^{{*}}}{\Vert {C^{-1}{} \mathbf{e}} \Vert _\infty }. \end{aligned}$$

On the other hand, the absolute value \(F_c \) of the critical force is evaluated as \(F_c =-\langle {\mathbf{F},\mathbf{e}} \rangle \), where \(\mathbf{F}\) is given by Eq. (B.2) with \(\delta \) being replaced by \(\delta _c \). In other words, we have [see also (96)]

$$\begin{aligned} F_c =\delta _c \langle {\mathbf{e},C^{-1}{} \mathbf{e}} \rangle . \end{aligned}$$

In order to estimate the pull-off force \(F_c \), we will make use of the following inequality [46]:

$$\begin{aligned} \langle {\mathbf{z},\mathbf{z}} \rangle ^{2}\le \langle {C\mathbf{z},\mathbf{z}} \rangle \langle {C^{-1}{} \mathbf{z},\mathbf{z}} \rangle . \end{aligned}$$

In particular, for the vector \(\mathbf{e}\), when \(\langle {\mathbf{e},\mathbf{e}} \rangle =N\), from (B.6) it follows that

$$\begin{aligned} \langle {\mathbf{e},C^{-1}{} \mathbf{e}} \rangle \ge \frac{N^{2}}{\langle {\mathbf{e},C\mathbf{e}} \rangle }. \end{aligned}$$

By the properties of the vector norm, we get

$$\begin{aligned} \langle {\mathbf{e},C\mathbf{e}} \rangle =\sum \limits _{k=1}^N {\sum \limits _{j=1}^N {C_{jk} } } \le N\Vert C \Vert _\infty . \end{aligned}$$

Thus, from (B.7) and (B.8), it follows that

$$\begin{aligned} \langle {\mathbf{e},C^{-1}{} \mathbf{e}} \rangle \ge \frac{N}{\Vert C \Vert _\infty }. \end{aligned}$$

Again by the norm’s property, we have \(\Vert {C^{-1}{} \mathbf{e}} \Vert _\infty \le \Vert {C^{-1}} \Vert _\infty \), and therefore, in view of Eq. (B.4), this inequality implies that

$$\begin{aligned} \delta _c \ge \frac{f^{{*}}}{\Vert {C^{-1}} \Vert _\infty }. \end{aligned}$$

Now, collecting formulas (B.5), (B.9), and (B.10), we obtain

$$\begin{aligned} F_c \ge \frac{Nf^{{*}}}{\Vert C \Vert _\infty \Vert {C^{-1}} \Vert _\infty }. \end{aligned}$$

Then, if \(R_*(C)>0\), where \(R_{*}(C)\) is defined by (97), the following inequality holds [47]:

$$\begin{aligned} \Vert {C^{-1}} \Vert _\infty \le \frac{1}{R_*(C)}. \end{aligned}$$

Therefore, from (B.11) and (B.12), we finally arrive at the lower estimate (98).

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Argatov, I., Li, Q. & Popov, V.L. Cluster of the Kendall-type adhesive microcontacts as a simple model for load sharing in bioinspired fibrillar adhesives. Arch Appl Mech 89, 1447–1472 (2019).

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  • Adhesive contact
  • Cluster of microcontacts
  • Load sharing