General Equations Describing Elastic Indentation Depth and Normal Contact Stiffness versus Load

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. Copyright 2000 Academic Press.     

On the time and indentation depth dependence of hardness,dissipation and stiffness in polydimethylsiloxane

Nanoindentation tests were performed on polydimethylsiloxane to characterize its mechanical behavior at different indentation depths and loading times. Astonishing indentation size effects have been observed in these experiments where the universal hardness increases by about 15 times from indentation depths of 5000 down to 100 nm. The hardness was found to depend on the loading time at small indentations, while at larger indentation depths the hardness hardly changed with loading time. In an attempt to unveil the underlying deformation mechanisms, an in-depth experimental study is pursued in this article with detailed analysis of the experimental data. Applying different loading times, the indentation experiments were evaluated at indentation depths from 100 to 5000 nm with respect to (a) universal hardness, (b) ratio of remaining indentation depth after unloading to maximum indentation depth, (c) ratio between elastic and total indentation works, and (d) indentation stiffness at maximum applied force. All these characteristics are found to be significantly different compared to a reference material that does not exhibit indentation size effects. The corresponding experimental data has been analyzed with an existing indentation depth dependent hardness model for polymers that has been motivated by a Frank elasticity related theory incorporating rotation gradients.     

Continuous stiffness mode nanoindentation response of poly(methyl methacrylate) surfaces

Indentation is a comparatively simple and virtually nondestructive method of determining mechanical properties of material surfaces by means of an indenter inducing a localized deformation. The paper present experimental results of the load-displacement curves, the hardness and the elastic modulus data, and associated analysis for poly(methyl methacrylate) (PMMA) surfaces as a function of contact displacement. The experimental results include continuous stiffness indentations performed using constant loading rate and constant displacement rate experiments. The continuous stiffness indentation involves continuous calculation of a material stiffness, and hence hardness and elastic modulus of surfaces, during discrete loading-unloading cycles, as in a conventional indentation routine, and in a comparatively smaller time constant. The dependence of the compliance curves, the hardness, the elastic modulus and the plasticity index upon the imposed penetration depth, the applied normal load and the deformation rate are described. Tip area and load frame calibrations for the continuous stiffness indentation are also reported. The paper includes practical considerations encountered during indentation of polymers specifically at low penetration depths. The experimental results show a peculiarly harder response of PMMA surfaces at the submicron (near to surface) layers.     

Length-scale dependence of variability in epoxy modulus extracted from composite prepreg

Property variability in conjunction with morphological variability are important sources of uncertainty in composite modeling. While image processing of experimental microstructures has enabled accurate quantification of morphological variability, the characterization of material variability is not as well established. In this study, the local material properties of epoxy extracted from a prepreg sheet was determined using nanoindentation with a spherical indenter tip with a radius of 50 μm. Indentations were carried out at four different indentation depths to evaluate the change in the variability of epoxy modulus with the sampling volume. For each length scale studied, 40 indentations were carried out to determine the variability in epoxy modulus. A significant decrease was observed in the coefficient of variation as the indentation depth increased. The corresponding modulus distributions were quantified. The results suggest that, similar to morphological variability, material variability is length-scale dependent and the appropriate variability associated with the selected length scale must be considered for stochastic modeling of composite structures.     

Measuring microelastic properties of stratum corneum

We explore the compression moduli of a thin biological tissue through probe microscopy. The elastic modulus (E') of isolated stratum corneum is measured at varying depths through the use of an atomic force microscope (AFM) as well as a nano-indentor (Hysitron Triboscope). In addition, a nano-DMA is used to measure visco-elastic properties. Measurements on dry and hydrated stratum corneum show an order of magnitude difference in E' and the measured tandelta (E'/E') is seen to increase from approximately 0.1 to 0.25. In addition, extensive validation of the experiments is conducted with different indentation probes at different force ranges to reveal the effects of indentor geometry and indentation depth on the measured elastic modulus. The sensitivity of the measurements is tested with applying known treatments to stratum corneum and exploring their effects on biomechanical parameters.     

Studies on mechanical and swelling behavior of polymer networks on the basis of the scaling concept. 3. The osmotic compressibility of gels

The osmotic compressibility of chemically crosslinked poly(vinyl acetate) (PVAc) gels in equilibrium with pure diluent has been determined by the method of decreasing equilibrium swelling in different swelling agents. The concentration dependence of the osmotic compressibility has been found to follow a scaling law with an exponent which was greater than that predicted by the mean-field theory. A comparison has been made between the elastic moduli determined by unidirectional deformation measurements and the compressional moduli obtained from the osmotic compressibility data.     

Atomic force microscopy measurement of the elastic properties of the kidney epithelial cells

Direct interaction force measurements using atomic force microscopy (AFM) were carried out between a silicon nitride tip and renal epithelial cells (Madin-Darby Canine Kidney-MDCK and proximal tubular epithelial cells derived from pig kidneys, LLC-PK1). The approaching (extending) portion of the force/distance curves is considered, and repulsive forces in the long range of 2-3 microm were seen in both MDCK as well as LLC-PK1 cells growing under normal conditions. The repulsive force in the shorter distance range of 50-200 nm was also observed, when cells were damaged exposing the underlying basal membrane. LLC-PK1 cells were more prone to damage than the MDCK cells, hence short-range forces were common in the former and long-range forces in the latter cells. The functional dependence of repulsive force on the indentation depth changes, at small indentation depth the force increases linearly, while at larger indentations the force is a quadratic function of the distance, which is attributed to the elasticity of the membrane and the solid-like response of cells, respectively. The oxalate treatment of cells for 2-4 h gives rise to an increase in the elastic modulus of the cells.     

Load and depth sensing indentation as a tool to monitor a gradient in the mechanical properties across a polymer coating: A study of physical and chemical aging effects

Load and depth sensing indentation has been used to characterize the elastic modulus and hardness of various polycarbonate films. This analytical technique is shown to be extremely suitable for the determination of gradients in these mechanical properties. Furthermore, it is demonstrated that such a gradient exists over a length of micrometers in chemically aged polycarbonate, but it is virtually absent in physically aged polycarbonate. From these results, it is concluded that, although the first 100 nm cannot be probed, physical aging occurs homogeneously throughout the bulk of the sample. However, chemical aging starts at the surface and moves progressively into the bulk of the material. From the study of these films, it appears that for the interpretation of these measurements, knowledge about the amount of creep occurring during the measurements and about the mechanical properties of the substrate on which these films are applied is needed. Creep can be measured with the same indenter through the application of a constant load for a period of time. Load and depth sensing indentation appears to be a powerful method for studying the physical and chemical aging of polymers. It is especially valuable for coatings and films for which conventional tensile testing is problematic. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1628–1639, 2004     

Scratch behavior of polycarbonate by Rockwell C diamond indenter under progressive loading

Rockwell C diamond indenter with a spherical tip of radius of 108 μm was used to slide on bulk polycarbonate under linearly increasing normal load from 5 mN to 20 N in order to investigate deformation and damage of ductile polymers during scratching. The peak value of the ratio of residual depth over penetration depth corresponds to a critical normal load, which can be used as the piecewise point for the subsection functions describing variation of tangential load, penetration depth and residual depth with normal load. The true adhesion interfacial friction coefficient was found to be 0.3 for contact between PC and diamond by three-dimensional model. Residual groove widths were found to be proportional to the square root of normal load, and provided useful information to characterize the radius of spherical tip. Values of fracture toughness of PC obtained by scratch-based methodologies under sufficiently large normal loads associated with brittle fracture and crack plane, are in excellent agreement with literature values. The material is under triaxial compressive stress state, making the flow and shear stresses much larger than the ones for uniaxial tensile test.     

Mechanics of nanoindentation on a monolayer of colloidal hollow nanoparticles

We explore the collective mechanical behavior of monolayer assemblies composed of close-packed arrays of hollow silica nanoparticles using a spherical nanoindentor. Seven types of well-defined hollow nanoparticles are studied with their radii ranging from 100 to 300 nm and shell thickness ranging from 14 to 44 nm. Micromechanical models reveal the underlying deformation mechanisms during indentation, where the consecutive contacting of the indentor with an increasing number of nanoparticles results in a nonlinear increase in the indentation force with penetration depth. Each contacted hollow nanoparticle successively locally bends, flattens, and then locally buckles. The effective indentation modulus of the monolayer film, which is obtained by a Hertzian fit to the experimental data, is found to be proportional to the elastic modulus of the nanoparticle shell material and scales exponentially with the ratio of particle shell thickness t to radius R to the power of 2.3. Furthermore, we find that for a constant film density with the same (t)/(R) of the constituent nanoparticles, smaller particles with a thinner shell can provide a higher effective indentation modulus, compared to their larger diameter and thicker shell counterparts. This study provides useful insights and guidance for constructing high-performance lightweight nanoparticle films and coatings with potential applications in tailoring stiffness and mechanical energy absorption.     

Size Effects in a Silica-Polystyrene Nanocomposite: Molecular Dynamics and Surface-enhanced Continuum Approaches

Size effects in a system composed of a polymer matrix with a single silica nanoparticle are studied using molecular dynamics and surface-enhanced continuum approaches. The dependence of the composite’s mechanical properties on the nanoparticle’s radius was examined. Mean values of the elastic moduli obtained using molecular dynamics were found to be lower than those of the polystyrene matrix alone. The surface-enhanced continuum theory produced a satisfactory fit of macroscopic stresses developing during relaxation due to the interface tension and uniaxial deformation. Neither analytical nor finite-element solutions correlated well with the size-effect in elastic moduli predicted by the molecular dynamics simulations.     

Viscoelastic characterization of polymers using instrumented indentation. I. Quasi‐static testing*

The use of instrumented indentation to characterize the mechanical response of polymeric materials was studied. A model based on contact between a rigid probe and a linear viscoelastic material was used to calculate values for the creep compliance and stress relaxation modulus for two glassy polymeric materials, epoxy and poly(methyl methacrylate), and two poly(dimethyl siloxane) (PDMS) elastomers. Results from bulk rheometry studies were used for comparison with the indentation stress relaxation results. For the two glassy polymers, the use of sharp pyramidal tips produced responses that were considerably more compliant (less stiff) than the rheometry values. Additional study of the deformation remaining in epoxy after indentation creep testing as a function of the creep hold time revealed that a large portion of the creep displacement measured was due to postyield flow. Indentation creep measurements of the epoxy with a rounded conical tip also produced nonlinear responses, but the creep compliance values appeared to approach linear viscoelastic values with decreasing creep force. Responses measured for the unfilled PDMS were mainly linear elastic, with the filled PDMS exhibiting some time‐dependent and slight nonlinear responses in both rheometry and indentation measurements. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 1794–1811, 2005     

Nanoindentation creep of nonlinear viscoelastic polypropylene

Uniaxial tensile creep tests at various applied stresses were carried out to demonstrate that PP is nonlinear viscoelastic. A novel phenomenological model consisting of springs, dashpots, stress-locks and sliders was proposed to describe the nonlinear viscoelasticity. Indentation creep tests at different applied load levels were also performed on nonlinear viscoelastic PP. It was found that the shear creep compliance varies with the applied load level when the applied load is less than 5 mN, which means the indentation creep behavior was nonlinear. To find the real reason for the nonlinearity in indentation creep tests, the elastic modulus at various indentation depths was measured using continuous stiffness measurements (CSM). By analyzing the variation of elastic modulus with indentation depth, the nonlinearity of indentation creep behavior was proved to be caused by the non-uniform properties in the surface of the specimen rather than nonlinear viscoelasticity.     

The effects of pile-up,viscoelasticity and hydrostatic stress on polymer matrix nanoindentation

It is well known that a clear disparity exists between the elastic modulus determined using macroscopic tensile testing of polymers and those determined using nanoindentation, with indentation moduli generally overestimating the elastic modulus significantly. The effects of pile-up, viscoelasticity and hydrostatic stress on the indentation modulus of an epoxy matrix material are investigated. An analysis of residual impressions using scanning probe microscopy indicates that material pile-up is insignificant. Viscous effects are negated by increasing the time on the sample during the loading/hold segment phases of the indentation test, and by calculating the contact stiffness at a drift-insensitive point of the unloading curve. Removing the effects of viscous deformation reduces the modulus by 10–13%, while also significantly improving the non-liner curve fitting procedure of the Oliver and Pharr method. The effect of hydrostatic stress on the indentation modulus is characterised using relations from literature, reducing the measured property by 16%. Once viscous and hydrostatic stress effects are accounted for, the indentation modulus of the material compares very well with the bulk tensile modulus, and modifications to standard indentation protocols for polymers are proposed.     

A comparison of JKR-based methods to analyze quasi-static and dynamic indentation force curves

The Johnson-Kendall-Roberts (JKR) theory of elastic contact, extended to take viscoelastic effects into account, is used to evaluate work of adhesion and modulus of elastomeric films. In this paper, we present a comparison of five approaches to analyze quasi-static and dynamic JKR force curve data obtained using instrumented indentation. The load-displacement experiments were performed using a 200-microm radius borosilicate glass sphere against poly(dimethyl siloxane) (PDMS). By applying a small oscillation to the tip during indentation, dynamic stiffness vs load data were also obtained for frequencies between 25 and 160 Hz. Direct curve fitting as well as simplified 2- and 3-point analysis methods were used to compare modulus values obtained from load-displacement and stiffness-load data. Fit methods not requiring determination of the initial point of tip-sample contact ("zero" displacement) provided modulus values closest to those obtained by direct curve fitting. The dynamic stiffness-load data revealed a frequency dependent modulus; load-displacement measurements obtained simultaneously were consistent with the relaxed, or low-frequency, modulus of the PDMS sample. These experiments demonstrate that both the frequency dependent and relaxed modulus can be obtained from a single experiment.     

Tailoring normal adhesion of arrays of thermoplastic, spring-like polymer nanorods by shaping nanorod tips

The tip shape of contact elements in hairy adhesion systems is crucial for proper contact formation and adhesion enhancement. While submicrometer terminal contact elements show much better adhesion performance than their larger counterparts, shaping their tips so as to maximize normal adhesion has remained challenging. We prepared durable nanorod arrays consisting of stiff, highly entangled thermoplastic polymers with rationally shaped tips by replication of anodic aluminum oxide (AAO). Nanorod arrays with pancake-like tips showed pronounced normal dry adhesion already for small loading forces. For small loading forces, adhesion forces significantly exceeded the loading forces. Both the absence of hysteresis in force/displacement curves and the pronounced durability of the nanorods in series of repeated attachment/detachment cycles suggest that the nanorods behave like elastic springs. Experimental load-adhesion curves were reproduced with a modified Schargott-Popov-Gorb (SPG) model, assuming that contacts between probe and individual nanorods are sequentially formed with increasing indentation depth.     

Hydrogen-bonded network of hydration water around model solutes

Clustering of water molecules in the hydration shells of spherical structureless solutes was studied in dependence on thermodynamic state, solute radius R(sp) and strength U(0) of water-solute interaction. Two qualitatively different clustering states of hydration water have been found: an "ordered" state with a hydrogen-bonded (H-bonded) network, which includes most of the hydration water, and a "disordered" state with small H-bonded clusters of hydration water. The transition from the ordered to disordered state occurs upon increasing temperature and decreasing pressure. This percolation transition is rounded due to the finite solute size and occurs in some temperature (pressure) interval. A finite-size scaling was applied to determine the transition temperature T(∞) in the limit R(sp)→∞. Strengthening of the water-solute interaction strongly enhances the stability of the ordered state: the transition temperature increases by about 35 °C, when U(0) decreases by 1 kcal mol(-1). At T > T(∞) and fixed U(0), the stability of the H-bonded water network increases upon decreasing solute size.     

Analysis of Solvent Effect on Mechanical Properties of Poly(ether ether ketone) Using Nano-indentation

Poly (ether ether ketone)(PEEK) is a high-performance semi-crystalline thermoplastic polymer.Exposure of the polymeric surface to solvents can have a strong effect like softening/swelling of polymeric network or dissolution.In this study, nano-indentation analysis was performed to study the effect of acetone on the surface mechanical properties of PEEK using different exposure time.The experiments were performed with a constant loading rate (10 nm/s) to a maximum indentation displacement (1000 nm).A 30-second hold segment was included at the maximum load to account for any creep effects followed by an unloading segment to 80% unloading.The indentation hardness and the elastic modulus were computed as a continuous function of the penetration displacement in the continuous stiffness mode (CSM) indentation.The experimental data showed that the peak load decreased from ~5.2 mN to ~1.7 mN as exposure time in solvent environment increased from 0 to 18 days.The elastic modulus and the hardness of PEEK samples also displayed a decreasing trend as a function of exposure time in the solvent environment.Two empirical models were used to fit the experimental data of hardness as a function of exposure time which showed a good agreement with the experimental values.     

Mechanical and electrical properties of the phosphor‐doped nano‐silicon film under an external electric field

The mechanical and electrical properties of the phosphor‐doped nano‐silicon film (nc‐Si:H) prepared by the plasma‐enhanced chemical vapor deposition (PECVD) method under electric field have been studied by Tribolab system, which is equipped with nano‐electrical contact resistance (ECR) tool. During indentation, different voltages and loads were applied. The topography of the sample surface was studied by atomic force microscopy (AFM). The experimental results show that the roughness of the film is 5.69 nm; the electric current was measured through the sample/indenter tip with different loads at a fixed voltage, and it increased nonlinearly during the indentation. The maximum current value depth was shallower than the maximum depth of each indent due to the plasticity of the film. When the loading speed is increased to 250 µN/s, the microcrack occurred on the film; the hardness (H) and elastic modulus (E) changed with the voltage applied both in open circuit and in short circuit case, which resulted in different values of H/E rate from 0.082 to 0.096. Copyright © 2009 John Wiley & Sons, Ltd.     

Molecular-dynamics studies of bending mechanical properties of empty and C60-filled carbon nanotubes under nanoindentation

This paper utilizes molecular-dynamics simulations to investigate the mechanical characteristics of a suspended (10, 10) single-walled carbon nanotube (SWCNT) during atomic force microscopy (AFM) nanoindentation at different temperatures. Spontaneous topological transition of the Stone-Wales (SW) defects is clearly observed in the indentation process. The present results indicate that under AFM-bending deformation, the mechanical properties of the SWCNT, e.g., the bending strength, are dependent on the wrapping angle. In addition, it is also found that the radial dependence of the reduced formation energy of the SW defects is reasonably insensitive only for the small tubes. However, for tube diameters greater than 2.4 nm [corresponding to the (18, 18) CNT], the SW defects tend to be more radius sensitive. The results indicate that the bending strength decreases significantly with increasing temperature. This study also investigates the variation in the mechanical properties of the nanotube with the density of C60 encapsulated within the nanotube at various temperatures. It is found that, at lower temperatures, the bending strength of the C60-filled nanotube increases with C60 density. However, the reverse tendency is observed at higher temperatures. Finally, the "sharpest tip" phenomena between the probe and the tube wall and the elastic recovery of the nanotube during the retraction process are also investigated.