A Fast Numerical Approach for Investigating Adhesion Strength in Fibrillar Structures: Impact of Buckling and Roughness
Abstract
:1. Introduction
2. Materials and Methods
2.1. Problem Statement and Theory
2.1.1. The Loading Process
2.1.2. The Unloading Process
2.1.3. Roughness
2.2. Numerical Computation
Algorithm 1 Loading Phase |
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- No Buckling Condition: In this state, a fibril is not buckled. Under this condition, the adhesion force remains as described in Equation (4).
- Low Buckling Condition: In the low buckling state, the adhesion force of the ith fibril decreases to 75% of its value in the unbuckled state. This reduction is based on experimental observations of fibril adhesion forces [26].
- High Buckling Condition: In contrast, under high buckling conditions, the adhesion force of the fibril is reduced to 25% of its unbuckled value. This adjustment reflects experimental findings that indicate significant decreases in adhesion forces under severe buckling conditions, with values approximating 30% in some studies [28,29]. For the purposes of simulation, a conservative estimate lower than the observed value is used.
Algorithm 2 Unloading Phase |
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Multiple Simulations and Normalization
- Maximum Adhesion Force Normalization: The maximum adhesion force is normalized against the adhesion force of a fibril array with no variance in fibril height or surface roughness of the counter surface. This normalization is performed using the single fibril adhesion value given by Equation (4) () and the number of fibrils (n) in the array. This approach allows for a clear distinction of the effects of height variance in the fibril array or roughness in the rigid counter surface on the adhesion strength.
- Number of Fibrils in Contact () and Number of Buckled Fibrils () Normalization: These metrics are normalized to the total number of fibrils in the array (n). This normalization enables clear observation of the impact of the pre–load factor on the number of fibrils in contact and buckled fibrils.
- Relative Displacement Normalization: The relative displacement of the counter surface () is normalized to the maximum elongation of the fibril, which is calculated assuming no variation in fibril height or counter surface roughness before contact loss. This maximum elongation occurs during the unloading phase due to the adhesive interaction between the fibril and the counter surface. The maximum elongation is determined using Equation (4) and the stiffness of the fibril () under ideal conditions, where the pre–load is below the critical value and there are no height variations or surface roughness. This normalization procedure ensures that the elongation of a rough fibril array is accurately related to the maximum deformation experienced by a smooth fibril array in contact with a smooth, rigid surface.
3. Results
3.1. Adhesion between a Rigid Surface and a Rough Fibril Array
3.1.1. Effect of Number of Fibrils on the Adhesive Strength
3.1.2. Effect of Work of Adhesion on the Adhesive Strength
3.1.3. Effect of Both Surfaces Being Rough on the Adhesive Strength
4. Discussion
5. Conclusions
- Effect of Roughness on Adhesion: The adhesion strength exhibits an exponential decay with increasing roughness parameters. This relationship is consistent across various pre–loads. This finding aligns with existing studies which show that the adhesion strength decreases in a similar manner with respect to roughness. The numerical model effectively captured the influence of surface roughness on adhesion strength.
- Role of Fibril Number: The number of fibrils in the array has a negligible effect on normalized adhesion strength when considering a normal distribution of fibril heights. This finding contrasts with previous studies that incorporated backing layer effects and structural coupling [33,42,47], indicating the importance of model assumptions in predicting adhesion behavior.
- Impact of Pre–Load on Adhesion Strength: Increasing the pre–load enhances the adhesion strength up to a certain threshold, beyond which the fibrils begin to buckle. The critical pre–load is directly related to the onset of buckling, and exceeding this value significantly reduces adhesion due to the instability introduced by buckling.
Supplementary Materials
Funding
Data Availability Statement
Conflicts of Interest
References
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Physical Parameter | Value |
---|---|
Young’s modulus of fibril, : | MPa |
Young’s modulus of rigid surface, : | 100 GPa |
Poisson ratio of fibril, : | 0.5 |
Poisson ratio of rigid surface, : | 0.3 |
The radius of fibril, R: | 100 m |
The height of fibril, : | 600 m |
The number of fibrils, n: | 50, 100, 2000 |
Work of adhesion, W: | 0.05, 0.25 J/m2 [38] |
Pre–load factor, : | 0.1:50 |
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Eray, T. A Fast Numerical Approach for Investigating Adhesion Strength in Fibrillar Structures: Impact of Buckling and Roughness. Lubricants 2024, 12, 294. https://doi.org/10.3390/lubricants12080294
Eray T. A Fast Numerical Approach for Investigating Adhesion Strength in Fibrillar Structures: Impact of Buckling and Roughness. Lubricants. 2024; 12(8):294. https://doi.org/10.3390/lubricants12080294
Chicago/Turabian StyleEray, Turgay. 2024. "A Fast Numerical Approach for Investigating Adhesion Strength in Fibrillar Structures: Impact of Buckling and Roughness" Lubricants 12, no. 8: 294. https://doi.org/10.3390/lubricants12080294