Flat‐punch indentation of viscoelastic material

The indentation of standard viscoelastic solids, that is, the three‐element viscoelastic material, by an axisymmetric, flat‐ended indenter has been investigated theoretically. Under the boundary conditions of flat‐punch indentation of a viscoelastic half‐space, the solutions of the equations of viscoelastic deformation are derived for the standard viscoelastic material. Their generality resides in their inclusion of compressible as well as incompressible solids. They cover the two transient situations: flat‐punch creep test and load‐relaxation test. In experimental tests of their applicability, nanoindentation and microindentation probes under creep and relaxation conditions yielded a modulus from 0.1 to 1.1 GPa and viscosity from 1 to 37 Gpa · s for a crosslinked glassy polyurethane coatings. For bulk polystyrene, the values vary from 1 to 2 GPa and from 20 to 40 Gpa · s, respectively. The analysis here provides a fundamental basis for probing elastic and viscous properties of coatings with nanoindentation or microindentation tests. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 10–22, 2000     

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     

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     

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     

Measurement of viscoelastic properties for polymers by nanoindentation

A new method has been proposed and verified to measure the viscoelastic properties of polymers by nanoindentation tests. With the mechanical response of load–displacement curves at different loading rates, the parameters of creep compliance and relaxation modulus are calculated through the viscoelastic contact model. Dynamic thermomechanical analysis (DMA) tests are conducted to compare the results by the proposed technique. The results show that the correlation coefficients between DMA tests and the new method are above 0.9 in the entire range, which verified the feasibility of the method. The loading curves fitted by the model are identical to the experimental curves within the discrete points and so it shows that this technique is more suitable for general linear viscoelastic materials. Numerical creep tests are carried out to examine the effectiveness of the proposed method by input the Prony series calculated by the three-element Maxwell model and the viscoelastic contact model. The good agreement shows that the proposed technique can be applied in practice.     

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     

Bonding of a viscoelastic periodic rough surface to a rigid layer

A theoretical analysis is performed for the study of the bonding of a viscoelastic rough surface to a rigid substrate. The mechanics of contact and adhesion are studied with the Dugdale–Barenblatt model for surface interaction. Exact solutions are obtained for arbitrary surface profiles and loading histories. Detailed solutions are given for a power‐law viscoelastic material. This solution is used to determine the time for the self‐bonding of surfaces (solid sintering under zero load). The time to self‐bonding is shown to be extremely sensitive to the aspect ratio of the asperities. A closed form expression is derived for the time needed to achieve full contact when the surfaces are compressed with a load that increases linearly with time. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 545–561, 2002; DOI 10.1002/polb.10113     

Nanofluid polyurethane/polyurea resins—thin films and clearcoats

Solvent‐free nanofluids with surface functional groups composed of tetraalkylammonium ionic liquid organic salts and isothiocyanate decorating nanosilica cores are reported that exhibit viscous flow and glass transitions. These nanofluids when combined with hexamethylene diisocyanate and isophorone diisocyanate–polypropyleneglycol prepolymers are coated and autocured in humid air to produce thin films and clearcoats. These new resins are characterized thermally and by nanoindentation. Relative to the air‐cured prepolymer control, low nanofluid loading (1–4% w/w) produces increased storage moduli and hardness and decreased loss factors. High loading (8 and 16%) results in softening with increased toughness. While the control exhibited marked viscoelastic recovery, this recovery was largely arrested in the nanocomposite coatings. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2013 , 51, 3439–3448     

Vinyl esters: Low cytotoxicity monomers for the fabrication of biocompatible 3D scaffolds by lithography based additive manufacturing

Lithography based additive manufacturing technologies (AMT) like stereolithography or digital light processing have become appealing methods for the fabrication of 3D cellular scaffolds for tissue engineering and regenerative medicine. To circumvent the use of (meth)acrylate‐based photopolymers, that suffer from skin irritation and sometimes cytotoxicity, new monomers based on vinyl esters were prepared. In vitro cytotoxicity studies with osteoblast‐like cells proofed that monomers based on vinyl esters are significantly less cytotoxic than (meth)acrylates. Photoreactivity was followed by photo‐differential scanning calorimetry and the mechanical properties of the photocured materials were screened by nanoindentation. Conversion rates and indentation moduli between those of acrylate and methacrylate references could be observed. Furthermore, osteoblast‐like cells were successfully seeded onto polymer specimens. Finally, we were able to print a 3D test structure out of a vinyl ester‐based formulation by μ‐SLA with a layer thickness of 50 μm. For in vivo testing of vinyl esters these 3D scaffolds were implanted into surgical defects of the distal femoral bone of adult New Zealand white rabbits. The obtained histological results approved the excellent biocompatibility of vinyl esters. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2009     

Nanoindentation studies of polyimide thin films with various internal linkages in the diamine component

Polyimide thin films were synthesized from 3,3′,4,4′‐biphenyltetracarboxylic acid dianhydride (BPDA) and four different diamines (p‐phenylene diamine, 4,4′‐oxydiphenylene diamine, 4,4′‐biphenylene diamine, and 4,4′‐sulfonyldiphenylene diamine). The nanoindentation behavior of the resulting polyimides, namely, poly(p‐phenylene biphenyltetracarboximide) (BPDA‐PDA), poly(4,4′‐biphenylene biphenyltetracarboximide) (BPDA‐BZ), poly(4,4′‐oxydiphenylene biphenyltetracarboximide) (BPDA‐ODA), and poly(4,4′‐sulfonyldiphenylene biphenyltetracarboximide) (BPDA‐DDS), were investigated. Also, the morphological properties were characterized with a prism coupler and wide‐angle X‐ray diffraction and were correlated to the nanoindentation studies. The nanoindentation behavior and hardness varied quite significantly, depending on the changes in the chemical and morphological structures. The hardness of the polyimide thin films increased in the following order: BPDA‐DDS < BPDA‐ODA < BPDA‐BZ < BPDA‐PDA. For all the polyimide thin films, except that of BPDA‐BZ, the hardness decreased with an increase in the load. The birefringence, a measure of the molecular in‐plane orientation, increased in the following order: BPDA‐DDS < BPDA‐ODA < BPDA‐PDA < BPDA‐BZ. The X‐ray diffraction studies revealed that the crystallinity of the polyimide thin films varied with the changes in the chemical structure. The studies showed that the indentation response with an applied load and the hardness by nanoindentation for the BPDA‐based polyimides were closely related to the morphological structure. The nanoindentation and birefringence results revealed that the mechanical properties of the polyimide thin films were dependent on the crystallinity, which arose because of the chain order along the chain axis and the molecular packing order. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 861–870, 2004     

Contact stiffness of initially stressed neo‐Hookean solids

This work, using the solution given by Dhaliwal and Singh, presents analytical expressions of the incremental stress and displacement fields for the axisymmetrical indentation of initially stressed, incompressible neo‐Hookean solids. A simple relation for the contact stiffness, contact area, elastic constants, and finite stretch can be obtained for the indentation by any rigid axisymmetric indenter, which can be described as a smooth function. The contact stiffness increases with the initial finite stretching; the finite stretching makes materials harder to deform. The results provide a basis for evaluating the effects of residual stresses on the nanoindentation of materials from the viewpoint of finite deformation. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 2513–2521, 2004     

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.     

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     

Effect of repetition nanoindentation of GaN epilayers on a‐axis sapphire substrates

In this work, gallium nitride (GaN) epilayers were deposited on a‐axis sapphire substrate by means of metal organic chemical vapor deposition (MOCVD). Berkovich nanoindentation was used to explore the repetition pressure‐induced impairment of the GaN film. The observation of load‐displacement vs stress‐strain curves concludes that basal slip is implicated in the deformation on the A plane GaN. The increase in the hardness (H) and elastic modulus (E) was determined from cyclic nanoindentation, and resulted in a crack due to the formation of incipient slip bands and/or the to‐and‐fro motion of mobile dislocation. It is indicated that the generation of individual dislocation and residual deformation of the GaN films are showed by CL mapping analysis. From the morphological studies, it is revealed that the crack was found by means of atomic force microscope (AFM) technique at nine loading/reloading cycles even after the indentation beyond the critical depth on the residual indentation impression. Copyright © 2010 John Wiley & Sons, Ltd.     

Apparent depth‐dependent modulus and hardness of polymers by nanoindentation: Investigation of surface detection error and pressure effects

Nanoindentation is a widely used technique to characterize the mechanical properties of polymeric materials at the nanoscale. Extreme surface stiffening has been reported for soft polymers such as poly(dimethylsiloxane) (PDMS) rubber. Our recent work [J. Polym. Sci. Part B Polym. Phys. 2017 , 55, 30–38] provided a quantitative model which demonstrates such extreme stiffening can be associated with experimental artifacts, for example, error in surface detection. In this work, we have further investigated the effect of surface detection error on the determination of mechanical properties by varying the sample modulus, instrument surface detection criterion, and probe geometry. We have examined materials having Young's moduli from ∼2 MPa (PDMS) to 3 GPa (polystyrene) using two different nanoindentation instruments (G200 and TI 950) which implement different surface detection methods. The results show that surface detection error can lead to apparent large stiffening. The errors are lower for the stiffer materials, but can still be significant if care is not taken to establish the range of the surface detection error in a particular experimental situation. We have also examined the effect of pressure beneath the probe on the nanoindentation‐determined modulus of polystyrene with different probe geometries. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018 , 56, 414–428     

Research on viscoelastic behavior of semi-crystalline polymers using instrumented indentation

In this study, the viscoelastic behavior of a polyamide 12 (PA12) polymer was evaluated using instrumented indentation technology based on a rheological model. The creep compliance and retardation spectra were obtained to analyze the viscoelastic response during the holding stage according to the rheological model under different preloading conditions. The results showed that the viscoelastic responses were independent of the indentation depth or load under step loading conditions. However, the creep compliance increases, and the peak intensity of the retardation spectrum decreases with a decrease in the preloading rate owing to the structural relaxation observed during the preloading stage under ramp loading conditions. Furthermore, softening dispersion can be completed during the loading stage under continuous stiffness measurement (CSM) conditions. As the preloading strain rate changes, the peak of the retardation spectrum gradually decreases until it disappears completely. Moreover, studies on indentation creep using the CSM method are challenging because of the complicated viscoelastic response observed during the preloading stage.     

On the Use of the Nanoindentation Unloading Curve to Measure the Young's Modulus of Polymers on a Nanometer Scale

Summary: The nanoindentation test is a fundamental tool to assess the link between morphology and mechanical properties. The preliminary results of a more exhaustive study about the applicability to polymers of the most used procedure to determine elastic modulus by indentation are reported in this short communication. A departure of the experimental conditions from the theoretical assumptions and results that give rise to the Oliver and Pharr analysis is shown to occur under a wide range of experimental conditions, with applied loads and penetration depths covering several orders of magnitude and using different indenter geometries. Unloading curves with exponents significantly larger than 2 are observed in disagreement with the contact mechanics approach used by Oliver and Pharr.

An AFM image obtained in non contact mode of an indentation induced by a sharp AFM tip with a maximum applied load of ca 1.2 µN on amorphous PET.     

Viscoelastic properties of nanosilica‐filled epoxy composites investigated by dynamic nanoindentation

The viscoelastic properties of the epoxy filled with silica nanoparicles have been investigated by dynamic nanoindentation and characterized by the storage modulus and loss tangent. The materials studied are neat epoxy and silica/epoxy composites with silica volume fraction of 1, 3, 6, 10, and 14 vol %, respectively. The silica nanoparticles with an average diameter of 25 nm are found to disperse homogeneously in the epoxy matrix. The effect of the particle content, force frequency, and penetration load on the viscoelastic behavior is studied and discussed. The comparison with traditional testing methods such as tension, bending, and DMTA is made. Besides, theoretical results by using micromechanics models are also obtained and compared with the experimental results. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 1030–1038, 2009     

Effect of surface tension on the relaxation of a viscoelastic half-space perturbed by a point load

We study the effect of surface tension on the flattening of a surface perturbed by a point load subsequent to its removal. The surface bounds an infinite isotropic linear viscoelastic incompressible half space. The point load is initially applied for a sufficiently long time so that the half space is fully relaxed before the load removal. An exact solution is obtained assuming small deformation. We then specialize our theory to the case of a standard viscoelastic solid. There is an initial reduction of the surface displacement immediately after load removal that is found to be directly proportional to the ratio of applied load to surface tension. This is followed by a temporal decay of the surface profile that depends only on the relaxation time and the long and short time moduli of the viscoelastic solid. Our work also provides the Green's function for a suddenly applied point load on the surface of a viscoelastic half space. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016, 54, 274–280     

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