Abstract
Contact force–indentation depth measurements in contact experiments involving compliant materials, such as polymers and gels, show a hysteresis loop whose size depends on the maximum indentation depth. This depth-dependent hysteresis (DDH) is not explained by classical contact mechanics theories and was believed to be due to effects such as material viscoelasticity, plasticity, surface polymer interdigitation, and moisture. It has been observed that the DDH energy loss initially increases and then decreases with roughness. A mechanics model based on the occurrence of adhesion and roughness related small-scale instabilities was presented by one of the authors for explaining DDH. However, that model only applies in the regime of infinitesimally small surface roughness, and consequently it does not capture the decrease in energy loss with surface roughness at the large roughness regime. We present a new mechanics model that applies in the regime of large surface roughness based on the Maugis–Dugdale theory of adhesive elastic contacts and Nayak’s theory of rough surfaces. The model captures the trend of decreasing energy loss with increasing roughness. It also captures the experimentally observed dependencies of energy loss on the maximum indentation depth, and material and surface properties.
Highlights
A clear understanding of adhesive contact mechanics is critical for spatially mapping out a material’s mechanical properties using, for example, nanoindentation- and contact mode atomic force microscopy (AFM)-based techniques[1,2]
Depth-dependent hysteresis has been observed in a number of other contact experiments, which span various length scales from μm to cm and involve different soft materials such as gelatin, PDMS, and poly(n-butyl acrylate) (PNBA)[7,8,9]
The gray dashed curves are the fit of the loading and unloading branches of the measured P–h data to the JKR theory. (b) A plot showing the variation of total energy loss as function of the root mean square (RMS) roughness in the experiments
Summary
As a consequence of this assumption, the total number of depth-dependent asperities can be computed as the product of the asperity density and the area of the nominal contact region that forms after the occurrence of the large-scale “pull-in” instability during the remainder of the contact cycle’s loading phase. Where Ac0 and Achmin are areas of the nominal contact region at the large-scale “pull-in” instability point (i.e., h = 0, marked as state ii in Fig. 1(b)) and at the maximum indentation depth (i.e., h = hmin, see Fig. 1(b)), respectively. Since the depth-dependent asperities are the total number of asperities contained in the nominal contact region formed after the occurrence of the large scale “pull-in” instability, eq (11) applies to the population of all depth-dependent asperities. Where η, ΔAc, and 〈ΔEmd〉 are, respectively, given by eqs (9), (10), and (13)
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