Papangelo 2018 - Adhesion power law indenter.pdf

In the present paper we investigate indentation of a power-law axisymmetric rigid probe in adhesive contact with a "thin layer" laying on a rigid foundation for both frictionless unbounded and bounded compressible cases. The investigation relies on the "thin layer" assumption proposed by Johnson, i.e. the layer thickness being much smaller than the radius of the contact area, and it makes use of the previous solutions proposed by Jaffar and Barber for the adhesiveless case. We give analytical predictions of the loading curves and provide indentation, load and contact radius at the pull-off. It is shown that the adhesive behavior is strongly affected by the indenter shape; nevertheless below a critical thickness of the layer (typically below 1 µm) the theoretical strength of the material is reached. This is in contrast with the Hertzian case, which has been shown to be insensitive to the layer thickness. Two cases are investigated, namely, the case of a free layer and the case of a compressible confined layer, the latter being more "efficient", as, due to Poisson effects, the same detachment force is reached with a smaller contact area. It is suggested that high sensitive micro-/nanoindentation tests may be performed using probes with different power law profiles for characterization of adhesive and elastic properties of micro-/nanolayers.

Widely used the Bulychev-Alekhin-Shorshorov relation for analyzing nanoindentation load-displacement data to determine elastic modulus of a thin specimen does not account for the size of specimen since the BASh relation is based on analytical solutions of the contact problems for an elastic half-space. In order to model the substrate effect, the unilateral contact problem for a spherical indenter pressed against an elastic layer on an elastic halfspace is analyzed for different types of boundary conditions imposed at the interface between the specimen and the substrate. Approximate (asymptotically exact) solutions are obtained in explicit form. The influence of the substrate effect on the incremental contact stiffness is described in terms of the asymptotic constants possessing information about the thickness of the specimen and depending on the relative stiffness of the substrate.

Acta Mechanica, 2020

Nanoindentation is employed to characterize the mechanical properties at the nanoscale. This paper considers the mechanical response of a nanoscale elastic layer on an elastic substrate that is indented by an adhesively bonded flat-ended rigid cylindrical punch. The complete Gurtin-Murdoch continuum model is employed to capture the size effects. The contact problem is analyzed by relating displacements of the contact region to contact stresses by a flexibility equation system, which is developed by discretizing the contact region into annular elements. The flexibility equation involves displacement influence functions corresponding to axisymmetric normal and radial surface ring loads applied on the layer-substrate system. The displacement influence functions are derived by using the Hankel integral transforms. Convergence and accuracy of the proposed solution scheme are verified by comparing with limiting cases such as the classical elasticity solution. Selected numerical results indicate that the substrate becomes stiffer and the elastic field is size-dependent due to the surface energy effects.

2012

It is of interest to measure the modulus of rigidity at small indentation depths for many systems, such as thin films, nanocomposites, biomaterials, etc. Depth-dependence of the rigidity modulus of homogeneous soft materials is broadly observed in nanoindentation experiments. Typically, the modulus reaches its bulk value only when the indentation depth becomes relatively large. Nature of this effect (we suggest to call this "skin-effect" for short) is not well understood. It is not even clear if this is a real effect or an artifact. Here we present the results of precise indentation measurements based on the use of atomic force microscopy (AFM), which suggest that the skin-effect may be an artifact. It can be eliminated, and the bulk modulus can be measured at nanometer indentations if one (a) takes into account adhesion between the indenter and surface of interest, and (b) operates mostly within the linear stress−strain regime. To demonstrate it, we used three AFM probes of well-defined geometry (radii of the apex were 22, 810, and 1030 nm) to study the indentation of three different polymers of the bulk rigidity of 0.6−0.7 GPa (polyurethanes) and 2.8 GPa (polystyrene). The obtained force−indentation curves were processed through the Oliver− Pharr, Hertz, Johnson−Kendall−Roberts (JKR) and Derjaguin−Muller−Toporov (DMT) models. We found that the skin-effect disappeared when using dull (810 and 1030 nm) probes and processing the force-deformation data with either of the adhesion models (JKR or DMT). Moreover, the measured moduli were independent of the indentation depth. The values of the rigidity modulus were very close to the bulk values starting from the indentations of 2−3 nm. Such a small indentation seems to be the smallest one for soft materials at which the bulk modulus has been reached. When using the sharp (22 nm) probe, we were not able to reach the bulk moduli up to the maximum possible indentation allowed by the instrument 90 nm. The other sources of possible error in the modulus measurements are discussed. We conclude that the skin-effect originates mainly at both nonlinearity of stress−strain relation (occurs when using excessively sharp probes) and if the probe−surface adhesion is not taken into account (like in Oliver−Pharr and Hertz models).

Mechanics of Materials, 2017

The classic version of the depth-sensing indentation techniques assumes the estimation of the elastic contact modulus of a material sample by measuring the slope (the contact stiffness) of the initial part of the unloading branch of the force-displacement curve. This approach assumes that the curve at loading reflects both elastic and plastic deformations of the material, while the unloading is taking place elastically. Therefore, neglecting the plastic deformations, one can assume that the structure of the material is the same at both branches and the assumptions of the Hertz-type contact theory are valid for the unloading branch. However, the contact problem for an elastic film attached to a substrate depends on the properties of the substrate. Hence, the film contact modulus is usually estimated by measuring the slopes of the initial unloading force-displacement curves obtained for different maximal values of indentation depth, and fitting the experimental points by various empirical analytical dimensionless functions of the ratio between the contact radius, a , and the layer thickness, t. Here, analytical analysis of contact problems for coated materials is performed. Both re-scaling and asymptotic techniques are employed. Asymptotic analysis of the contact at the small-scale indentation range (the ratio a / t is small) shows that the formula of the contact stiffness derived for an elastic half-space, has to be multiplied by the socalled indentation scaling factor that is a function of a / t. Thus, the asymptotic approach allows us to take into account analytically the substrate effect. The analytical fitting function obtained agrees with both some known semi-empirical functional forms and the published experimental results on depth-sensing nanoindentation of thin metallic layers, while the function is in a disagreement with results obtained for inhomogeneous films of brittle materials such as coals. It is argued that the disagreement is caused by structural transformations (crushing) of the coals during loading.

International Journal of Solids and Structures, 2009

This paper examines the frictionless adhesive elastic contact problem of a rigid sphere indenting a thin film deposited on a substrate. The result is then used to model the elastic phase of micro-nanoscale indentation tests performed to determine the mechanical properties of coatings and films. We investigate the elastic response including the effects of adhesion, which, as the scale decreases to the nano level, become an important issue. In this paper, we extend the Johnson-Kendall-Roberts, Derjaguin-Muller-Toporov, and Maugis-Dugdale half-space adhesion models to the case of a finite thickness elastic film coated on an elastic substrate. We propose a simplified model based on the assumption that the pressure distribution is that of the corresponding half-space models; in doing so, we investigate the contact radius/film thickness ratio in a range where it is usually assumed the half-space model. We obtain an analytical solution for the elastic response that is useful for evaluating the effects of the film-thickness, the interface film-substrate conditions, and the adhesion forces. This study provides a guideline for selecting the appropriate film thickness and substrate to determine the elastic constants of film in the indentation tests.

Recent Patents on Nanotechnology, 2012

In the last years, Nanoindentation or Instrumented Indentation Technique has become a powerful tool to study the mechanical properties at micro/nanometric scale (commonly known as hardness, elastic modulus and the stress-strain curve). In this review, the different contact mechanisms (elastic and elasto-plastic) are discussed, the recent patents for each mechanism (elastic and elasto-plastic) are summarized in detail, and the basic equations employed to know the mechanical behaviour for brittle and ductile materials are described.

Surface & Coatings Technology, 2003

Local-probe instruments such as surface force apparatus or atomic force microscope are routinely used to extract the mechanical properties of thin-layers by subjecting them to indentation loading. Inadvertently, at these micro-or nano-indentation loads, the adhesive surface forces operating between the indenter and the thin-layer will contribute to the deformation. The well-established Johnson-Kendal-Roberts (JKR) theory is applied to extract the surface energy of the contacting solids. For this thin-layer system, the JKR theory should be an error as it is based on the indentation of an elastic half-space with an elastic sphere. A continuum mechanics approach coupled with non-dimensional analysis and finite elements simulation is presented to explain the indentation process of multi-layer elastic systems under the presence of adhesion. The emphasis is on spherical and flat punch probes. Computations of contact size and contact stiffness as a function of load are presented for a range of values of adhesion energies. ᮊ

Journal of Colloid and Interface Science, 2000

Continuum mechanics models describing the contact between two adhesive elastic spheres, such as the JKR and DMT models, provide a relationship between the elastic indentation depth and the normal load, but the general intermediate case between these two limiting cases requires a more complex analysis. The Maugis-Dugdale theory gives analytical solutions, but they are difficult to use when comparing to experimental data such as those obtained by scanning force microscopy. In this paper we propose a generalized equation between elastic indentation depth and load that approximates Maugis' solution very closely. If the normal contact stiffness can be described as the force gradient, that is the case of the force modulation microcopy, then a generalized equation between normal contact stiffness and load can be deduced. Both general equations can be easily fit to experimental data, and then interfacial energy and elastic modulus of the contact can be determined if the radius of the indenting sphere is known. C 2000 Academic Press

Applied Physics Letters, 2006

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