Viscoelastic modeling of nanoindentation experiments: A multicurve method

Nanoindentation is an increasingly used method of extracting surface mechanical properties of viscoelastic materials, especially polymers. Recently, Hutcheson and McKenna used a viscoelastic contact mechanics model to analyze the contact problem between a nanosphere and polystyrene surface. In nanoindentation experiments, the ramp loading test is a similar problem to the particle embedment experiment except that the indentation load function differs. The motivation in this work is to expand the Hutcheson and McKenna analysis to the nanoindentation problem. In particular, we illustrate the limitations of analyzing only a single load‐indentation curve, which does not provide enough information to determine the full range of the viscoelastic response of a polymer, and we show that performing a test sequence that includes multiple loading rates or indentation rates spanning two or more orders of magnitude greatly improves the extracted viscoelastic properties. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2014 , 52, 633–639     

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     

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     

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 extreme depth dependence of the hardness of PDMS rubber: A problem of false surface detection

There is significant interest in nanoindentation of materials yet mixed results for material properties have been reported, including modest depth dependence of the surface stiffness in metals and other crystalline materials and polymer glasses, as well as stiffening of several orders of magnitude in some studies of soft materials such as poly(dimethylsiloxane) (PDMS) rubber. At the same time, there are reports that suggest that the observed extreme stiffening in soft materials might be an artifact, and that such materials at most exhibit only mild stiffening. Unfortunately, a quantitative model of potential artifacts has not been provided. In the present work, we examine the problem of one potential artifact in the testing of soft materials, that of the difficulty of detecting the surface or “true zero” in the indentation test. We provide a quantitative estimate of the effect of error in surface detection on the measured force–displacement curves for the Berkovich tip geometry and find that the observed apparent stiffening is in agreement with our analysis. The significance of the results for testing of soft materials by nanoindentation is discussed. It is also shown that for hard/stiff materials the induced errors are smaller, but may still be significant in some circumstances. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017 , 55, 30–38     

Temperature dependent relaxation of a “solid–liquid”

We have used the Interfacial Force Microscope to perform temperature dependent indentation measurements on a model viscoelastic material, Silly Putty. By transforming time dependent stress relaxations into frequency dependent modulus, we can identify the temperature dependence of the elastic and viscous response of an experimentally challenging material. This technique promises to be useful in determining the mechanical properties of composite materials with microscopic spatial resolution. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 1285–1290, 2009     

Effect of large deformation and material nonlinearity on the JKR (Johnson–Kendall–Roberts) test of soft elastic materials

We study in detail the effect of large deformation and material nonlinearity on the JKR (Johnson–Kendall–Roberts) theory of adhesive contact for two systems. The first is a Neo‐Hookean hemisphere in adhesive contact with a smooth rigid substrate. The second is a smooth rigid spherical indenter in adhesive contact with a Neo‐Hookean half space. We show that our results are special cases of a general theory that models large deformation adhesive contact of spherical lenses. This theory shows that the solution of any large deformation JKR (LDJKR) problem can be obtained from the solution of a corresponding large deformation Hertz (LDH) problem. Using this theory, we extend the small strain JKR theory to the large deformation regime, the only restriction being that the materials are nonlinear elastic or hyperelastic. The adhesive contact problem for the two systems is solved using two methods. In method one, the LDJKR theory is obtained using finite element simulation results for a corresponding LDH problem; in method two, we solve the adhesive contact problems directly using a cohesive zone model to quantify adhesive interaction. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2912–2922, 2006     

Completely biodegradable composites of poly(propylene carbonate) and short,lignocellulose fiber Hildegardia populifolia

The composites of biodegradable poly(propylene carbonate) (PPC) reinforced with short Hildegardia populifolia natural fiber were prepared by melt mixing followed by compression molding. The mechanical properties, thermal properties, and morphologies of the composites were studied via static and dynamic mechanical measurements, thermogravimetric analysis, and scanning electron microscopy (SEM) techniques, respectively. Static tensile tests showed that the stiffness and tensile strength of the composites increased with an increasing fiber content. However, the elongation at break and the energy to break decreased dramatically with the addition of short fiber. The relationship between the experimental results and the compatibility or interaction between the PPC matrix and fiber was correlated. SEM observations indicated good interfacial contact between the short fiber and PPC matrix. Thermogravimetric analysis revealed that the introduction of short Hildegardia populifolia fiber led to a slightly improved thermooxidative stability of PPC. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 666–675, 2004     

AFM nanoindentation of viscoelastic materials with large end‐radius probes

We used atomic force microscopy (AFM) nanoindentation to measure mechanical properties of polymers. Although AFM is generally acknowledged as a high‐resolution imaging tool, accurate quantification of AFM nanoindentation results is challenging. Two main challenges are determination of the projected area for objects as small as AFM tips and use of appropriate analysis methods for viscoelastic materials. We report significant accuracy improvements for modulus measurements when large end‐radius tips with appropriate cantilever stiffnesses are used for indentation. Using this approach, the instantaneous elastic modulus of four polymers we studied was measured within 30 to 40% of Dynamic Mechanical Analysis (DMA) results. The probes can, despite their size and very high stiffnesses, be used for imaging of very small domains in heterogeneous materials. For viscoelastic materials, we developed an AFM creep test to determine the instantaneous elastic modulus. The AFM method allows application of a nearly perfect stepload that facilitates data analysis based on hereditary integrals. Results for three polymers suggest that the observed creep in the materials has a strong plastic flow component even at small loads. In this respect, the spherical indenter tips behave like “sharp” indenters used in indentation studies with instrumented indenters. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 1573–1587, 2009     

Wear and friction in glassy polymers: Microscratch on blends of polystyrene and poly (2,6-dimethyl-1,4-phenylene oxide)

The microscopic process of abrasive wear and friction in glassy polymers was studied by using a special microscratch technique. A miscible blend of polystyrene (PS) and poly(phenylene oxide) (PPO) was used. It was found that as the composition varies there seems to exist two wear regimes in the blends controlled by different breakdown mechanisms corresponding to the brittle—ductile transition. Detailed study of the contact loads and SEM micrographs indicate that abrasive wear in the glassy polymers is controlled by microcracking under the asperity contacts. The critical load τ_c for initiating microscopic cracks can be linked to the macroscopic wear via a statistical Weibull model where τ_c is taken to be the mean of a strength distribution function. On the other hand, the friction coefficient was found to be independent of the composition but to vary strongly with the contact load. It approaches zero at the extrapolated zero load, but increases rapidly and eventually levels off with contact load. This behavior can be understood by a simple frictional adhesion model in which the polymer deformation during a frictional contact is analyzed by considering the compressive plastic ploughing and shearing yielding around the asperity contact. The shear strength S_o of the polymer/asperity contacts was found to vary with the normal load. The vertical scratch hardness H_v, which characterizes the spontaneous indentation yielding on the polymer surface, was found to be independent of scratch length and depth, and indeed can be regarded as a material constant. Although both S_o and H_v can accurately describe the frictional behavior of the glassy polymers, they bear no correlation to abrasive wear in the same materials. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 1295–1309, 1997     

A contact mechanics method for characterizing the elastic properties and permeability of gels

When a saturated gel immersed in the same liquid is suddenly brought into contact with a smooth rigid indenter, the liquid cannot immediately flow out of the pores, and so the gel initially behaves as an incompressible material. This gives rise to a pressure gradient in the liquid phase and the liquid flows until the pressure in it goes to zero everywhere, and all the stresses are transferred to the elastic network. As a result of the flow, the force needed to maintain a constant contact area relaxes with time. In this work, we study the feasibility of using an indentation test to measure this time‐dependent force and to determine the elastic modulus, the Poisson's ratio, and the permeability, D_p, of the network. Specifically, we consider a two‐dimensional Hertz contact problem of a rigid circular cylinder indenting on a half space consisting of an elastic gel. The network of the gel is assumed to be linearly elastic and isotropic, and liquid flow within the gel is assumed to obey Darcy's law, which states that the flux is proportional to the pressure gradient. Exact expressions are obtained for the initial and final force required to maintain a given contact length. These expressions allow us to determine the elastic constants of the network. The permeability of the network can be obtained from the time‐dependent relaxation of the load, which is obtained by solving the exact continuum equations. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 359–370, 2006     

Click chemistry in materials synthesis. 1. Adhesive polymers from copper‐catalyzed azide‐alkyne cycloaddition

The copper(I)‐catalyzed cycloaddition reaction between azides and alkynes has been employed to make metal‐adhesive materials. Copper and brass surfaces supply the necessary catalytic Cu ions, and thus the polymerization process occurs selectively on these metals in the absence of added catalysts. Alternatively, copper compounds can be added to monomer mixtures and then introduced to reducing metal surfaces such as zinc to initiate polymerization. The resulting materials were found to possess comparable or superior adhesive strength to standard commercial glues, and structure‐activity correlations have identified several important properties of the monomers. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 4392–4403, 2004     

Depth-sensing indentation applied to polymers: A comparison between standard methods of analysis in relation to the nature of the materials

Mechanical data (hardness and elastic modulus) from instrumented indentation testing are often extracted assuming linear elasticity in the initial portion of the unloading. The method is nowadays widely accepted as a convenient tool to interpret depth-sensing data, however it is a matter of controversy when applied to polymer materials due to their time-dependent behavior. More recently, Loubet and co-workers applied continuous stiffness measurements (CSM), consisting of superimposing a small oscillation to the quasi-static component of loading, to the study of the mechanical properties of polymers and proposed a new model to account for the apparent increase in the contact area detected at the first stages of contact. The present work offers a comparative study between the Loubet’s model using CSM and the procedure yielding a single reading from the onset of unloading. A wide range of thermoplastic polymer materials including glassy and semicrystalline polymers have been investigated. The most important equations employed for each method are summarized and the advantages and disadvantages of employing one procedure or the other are discussed. The differences found between the results obtained from both approaches are discussed in relation to the nature of the polymer material. A comparison between mechanical data extracted from indentation measurements and from classical dynamic mechanical analysis is offered.     

Micromechanical modeling of the tensile behavior of oriented polyethylene

The stacked lamellar morphology commonly found in extruded semicrystalline materials has a strong influence on the flow direction, with respect to the loading direction, and on the stability and localization phenomena in tensile experiments. A multiscale numerical model was used to simulate the effect on the macroscopic behavior of a stacked lamellar microstructure. The model established a link between the microscopic, the mesoscopic, and the macroscopic levels. The constitutive properties of the material were identified for the crystallographic and amorphous domains. The average fields of an aggregate of individual phases, having preferential orientations, formed the constitutive behavior of the extruded material. The microscopic morphology of the extruded high‐density polyethylene is based on wide‐angle X‐ray diffraction experiments. The macrostructure was described by a finite element model. The microstructure‐induced deformation hardening in the extrusion direction was found to stabilize the macrostructure when it was loaded in the flow direction. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 2983–2994, 2004     

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.     

Détermination du module d'young,de la dureté et de la contrainte d'écoulement plastique de dépôts par nano-indentation

Variable residual stresses are present in the metallic coatings of tungsten and chromium manufactured by PVD and modify the quantities measured by depth-sensing indentation instruments: depth of penetration and contact stiffness for a given load. By considering the results of experiments and numerical simulations we discuss the corrections required for improving the determination of the mechanical characteristics of such coatings.     

Mechanics of contact and adhesion between viscoelastic spheres: An analysis of hysteresis during loading and unloading

Detailed finite element simulations are carried out to study the adhesive contact of viscoelastic spheres. The spheres are brought into contact by a compressive force that increases at a constant rate. Upon reaching a maximum load, the spheres are unloaded until they separate. We studied in detail the effect of loading and unloading rates on hysteresis and on the pull‐off force for a standard viscoelastic solid. The surface interaction is modeled by the Dugdale–Barenblatt model. Numerical results are compared with analytical models for bonding and debonding, including a recent theory proposed by Johnson. There is excellent agreement between analytical and finite element results for the bonding phase. However, for the debonding phase, current analytical models break down unless the loading and unloading rates are slow in comparison with the material relaxation time. Based on the finite element results, a simple approximate analytical model is proposed to quantify adhesive contact in the debonding phase. We also examine the dependence of hysteresis on interfacial parameters such as the cohesive strength and the intrinsic work of adhesion. Our results show that viscoelastic adhesive contact depends on the details of the surface interaction and cannot be determined solely by the work of adhesion. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 772–793, 2002     

Side‐chain functionalized polymers containing bipyridine coordination sites: Polymerization and metal‐coordination studies

Monomers containing (trisbipyridine) ruthenium(II), (bisbipyridine) palladium(II), and heteroleptic ruthenium complexes were synthesized and polymerized via ruthenium‐catalyzed ring‐opening metathesis polymerization in nonpolar solvents. The solubility of the resulting polyelectrolytes in nonpolar solutions could be tuned by alkyl functionalization of the ligands around the metal centers. These polymers are the first polynorbornenes containing a 2,2′‐bipyridine‐based metal complex at each repeating unit and might be used in numerous applications, including luminescent and electroluminescent materials. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 2973–2984, 2004     

The mechanics of tack: Viscoelastic contact on a rough surface

This work considers the mechanics of tack in viscoelastic materials. We study a particular tack test in which a flat, rigid probe is brought into contact with the rough surface of a viscoelastic material. The rough surface is modeled as consisting of many spherical asperities of varying heights but all having the same radius. Because of the asperities, the apparent contact area can be much greater than the actual contact area, which is regarded as the key parameter that controls tack. We show how this actual contact area evolves with time under different loading conditions. Our formulation is different from previous theories in that it explicitly accounts for the fact that asperities of different heights are subjected to different loading histories. Explicit solutions are given for the cases of a constant load test, a load relaxation test, and a constant displacement rate test. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1485–1495, 2000