Gecko Foot

In nature, various species apply advanced techniques for specialized tasks. For instance, gecko lizards use dry adhesion forces such as van der Waals forces to climb walls. Dry adhesion is a phenomenon of interest for sticking because it requires no energy to hold on, and no residue is left on the surface. Gecko lizards have inspired researchers to develop synthetic gecko foot hairs for use in, for example, robots.

A strand of hair on a gecko foot is a very complex biological structure with hierarchical nanosections and microsections. On its feet, a gecko has billions of nanoscale hairs that are in contact with surfaces while it climbs. These nanohairs are attached to microscale hairs, which are on the tip of the gecko’s toes.

Critical design parameters for nanohairs to achieve optimal sticking are hair length, detach angle, distance between nanohairs, and the cross-sectional area of a single strand of hair. By varying these parameters, it is possible to design hairs that can stick to very rough surfaces. At the same time, they must be stiff enough to avoid sticking to each other. Proper material choices help achieving the design goals while providing the required adhesion force. Typically the Young’s modulus for materials used in synthetic nanohair vary between 1 GPa and 15 GPa.

Model Definition

This model contains the hierarchy of synthetic gecko foot hair where nanoscale and microscale cantilever beams describe the seta and spatula parts of a spatular stalk attached to a gecko foot. The basis of the analyzed structure is a microscale stalk with the following dimensions: width, 4.53 μm; height, 4.33 μm; and length, 75 μm. At the end of the microhair, 169 nanohairs are attached and they have dimensions of 0.18 μm, 0.17 μm, and 3 μm, respectively. The microhair is fixed at the far end, while the contact and friction forces appear as surface loads at the end of each nanohair. The free-body diagram of a micro/nanohair in Figure 1 illustrates the applied forces, which are set to 0.4 μN for the contact force and 0.2 μN for the friction force with 60° contact angle to target surface. The structure is made of β-keratin with a Young’s modulus of 2 GPa and a Poisson’s ratio of 0.4. The model was inspired by Ref. 1.

Figure 1: Model of the tip of a gecko foot hair.

Results and Discussion

The plot in Figure 2 shows the von Mises equivalent stress in the model. Table 1 lists the maximum values of the von Mises stress, total displacement, and principal strain:

Table 1: Stress, displacement, and principal strain results ignoring geometric nonlinearity.
Maximum von Mises Stress (N/m2)	Maximum Total Displacement (μm)	Maximum First Principal Strain
9.52·107	12.6	0.0505

The maximum von Mises stress in the analyzed model is almost twice the value of the material’s yield stress, which clearly indicates that further investigation is required.

The deformed plot in Figure 2 shows that displacement is large, while the results in the table above indicate that the maximum strain is moderate. Hence, the next step is to enable the geometric nonlinearity within the linear elastic material model. (An alternative would be to search for a suitable hyperelastic material model, which would require the corresponding material data.)

Figure 2: Deformed shape plot of the von Mises stress in a synthetic gecko foot ignoring geometric nonlinearity. The plot shows the displacements without any scaling.

Figure 3: The result recomputed with the geometric nonlinearity taken into account.

You include the geometric nonlinearity by selecting the corresponding check box in the Linear Elastic Material Model feature node. The resulting changes in the model equations are shown under the Equation section therein.

The plot in Figure 3 shows the resulting von Mises stress distribution and the deformation, and Table 2 below gives the recomputed maximum values.

Table 2: Stress, displacement, and principal strain results including geometric nonlinearity effects.
Maximum von Mises Stress (N/m2)	Maximum Total Displacement (μm)	Maximum First Principal Strain
1.73·107	2.93	8.85·10-3

The results show that without the geometric nonlinearity taken into account, the model overpredicts the maximum von Mises stress by more than a factor of five, and the maximum displacement by more than a factor of four. Furthermore, the maximum strain computed with the geometric nonlinearity included becomes less than 1%, which eliminates the need of further analysis involving more complicated hyperelastic material models.

References

1. G. Shah and I. Lee, Finite Element Analysis of Gecko Foot Hairs for Dry Adhesive Design and Fabrication, Dept. of Mechanical Engineering — NanoRobotics Lab, Carnegie Mellon Univ., Pittsburgh.

2. M. Sitti and R.S. Fearing, “Synthetic Gecko Foot-Hair Micro/Nano-Structures for Future Wall-Climbing Robots”, Proc. IEEE Robotics and Automation Conf., Sept. 2003.

3. J. Vincent, Structural Biomaterials, rev. ed., Princeton University Press, 1990.

MEMS_Module/Actuators/gecko_foot

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Fc	0.4[uN]	4E-7 N	Contact force
Ff	0.2[uN]	2E-7 N	Friction force
theta	pi/3	1.0472	Contact angle
Dm	75[um]	7.5E-5 m	Microhair length
Hm	4.33[um]	4.33E-6 m	Microhair width
Wm	4.53[um]	4.53E-6 m	Microhair height
Dn	3[um]	3E-6 m	Nanohair length
Hn	0.17[um]	1.7E-7 m	Nanohair width
Wn	0.18[um]	1.8E-7 m	Nanohair height
Area	Wn*Hn	3.06E-14 m²	Cross-sectional area of the spatulas

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Nanohair

In the toolbar, click

In the window for , locate the section.

In the text field, type Wn.

In the text field, type Dn.

In the text field, type Hn.

Locate the section. In the text field, type -Dn.

In the text field, type Nanohair.

Next, add object selections for the nanohair’s root and end. An object selection is defined by a set of geometric entities at a specified level selected from the geometric objects that precede the object selection feature in the geometry sequence. Their main advantage compared to regular selection features is that object selections propagate through the geometry sequence, which make them robust and convenient to apply in cases like the present one.

Nanohair ends

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

On the object , select Boundary 3 only.

In the text field, type Nanohair ends.

Nanohair roots

In the toolbar, click

and choose .

Click the

button in the toolbar.

In the window for , locate the section.

From the list, choose .

On the object , select Boundary 6 only.

Click the

button in the toolbar.

Click

In the text field, type Nanohair roots.

Array 1 (arr1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

In the text field, type 13.

Locate the section. In the text field, type (Wm-Wn)/12.

In the text field, type (Hm-Hn)/12.

Click

Click the

button in the toolbar.

Microhair

In the toolbar, click

In the window for , locate the section.

In the text field, type Wm.

In the text field, type Dm.

In the text field, type Hm.

Click

In the text field, type Microhair.

Microhair end

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

On the object , select Boundary 3 only.

It might be easier to select the correct boundary by using the window. To open this window, in the toolbar click and choose . (If you are running the cross-platform desktop, you find in the main menu.)

Click

In the text field, type Microhair end.

Click the

button in the toolbar.

Form Union (fin)

Use an assembly to connect parts of the geometry of significantly different dimensions.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Clear the check box.

Click

The model geometry is now complete.

Definitions

Because of the use of assembly, you need to add an identity pair to set up the displacement field continuity at the interfaces, where the nanohairs are attached to the microhair. Configure the identity to operate on the material frame, because this is the frame used within the interface.

Identity Boundary Pair 1 (p1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Click to select the

toggle button.

From the list, choose .

Define a rotated coordinate system to account for the contact angle.

Rotated System 2 (sys2)

In the toolbar, click

and choose .

In the window for , locate the section.

Find the subsection. In the β text field, type pi/2-theta.

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	2e9	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.4	1	Young’s modulus and Poisson’s ratio
Density	rho	1200	kg/m³	Basic

Definitions

Set up a maximum operator to compute maximum values of the von Mises stress, displacement magnitude, and principal strain over the entire geometry.

Maximum 1 (maxop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Variables 1

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
max_v_Mises	maxop1(solid.mises)	N/m²	Maximum von Mises stress
max_disp	maxop1(solid.disp)	m	Maximum displacement magnitude
max_ep1	maxop1(solid.ep1)		Maximum first principal strain