Indentation responses of piezoelectric films

Frictionless indentation responses of transversely isotropic piezoelectric film/rigid substrate systems under circular cylindrical indenter (i.e., punch), conical indenter (i.e., cone), and spherical indenter (i.e., sphere) are investigated. Both insulating and conducting indenters are considered. The technique of Hankel transformation is employed to derive the corresponding dual integral equations for the mixed boundary value indentation problems. For the two limiting cases of infinitely thick and infinitely thin piezoelectric films, closed-form solutions are obtained. For piezoelectric films of finite thickness, a numerical method is constructed to solve the dual integral equations and semi-empirical models having only two unknown parameters are proposed for the responses of indentation force, electric charge and electric potential, and contact radius. With the two parameters inferred from the numerical results, the semi-empirical formulae are found to provide good estimates of the indentation responses for the two limiting cases of infinitely thick and thin piezoelectric films, as well as those in between. The inferred parameters in the proposed semi-empirical formulae for normalized indentation force and electric charge are checked against four different piezoelectric materials and are found to be insensitive to the selection of piezoelectric materials. It is believed that the proposed semi-empirical indentation formulae are useful in developing experimental indentation techniques to extract the material properties of piezoelectric films.     

Investigation of Structural Processes During Indentation of Polymers by Synchrotron Radiation Microdiffraction

Abstract

A Vickers microindentation setup has been integrated into the scanning setup used at the European Synchrotron Radiation Facility (ESRF) microfocus beamline (ID 13). Ex situ wide‐angle x‐ray scattering experiments performed on an isotactic polypropylene (iPP) single fiber show changes in chain orientation and breakup of the monoclinic α‐modification into domains due to plastic deformation in the indented zone. The systematic mapping of the perturbed zone allows spatially dependant structural changes to be determined. Similar experiments for a Vectra single fiber show a partial transformation of the pseudohexagonal phase into the orthorhombic phase. In situ experiments allow following structural changes as a function of time and applied force. During indentation of an iPP fiber, domain doubling during indentation and single domain recovery after the tip has been retracted is observed.     

Length scale effects in epoxy: The dependence of elastic moduli measurements on spherical indenter tip radius

Significant increases in the measured elastic moduli with decreasing indentation depth have been previously found in various polymers by indentation tests with a Berkovich tip at micro-to nanometer length scales. These increases in the determined elastic moduli were related to second order displacement gradients which increase with decreasing depth when a conical tip is applied. When a spherical tip is applied, such depth dependence should not be present as the second order displacement gradients remain essentially unchanged with indentation depth. However, these gradients should be proportional to the radius of the spherical tip. To examine the notion of second order displacement gradient dependence in measurements of elastic moduli, indentation experiments are conducted on epoxy with spherical tips of different nominal radii. Accounting for tip imperfections, an increase in the determined elastic moduli is found with decreasing tip radius, which corroborates the notion of second order displacement gradient dependence.     

Nanoindentation and the indentation size effect: Kinetics of deformation and strain gradient plasticity

A study of the indentation size effect (ISE) in aluminum and alpha brass is presented. The study employs rate effects to examine the fundamental mechanisms responsible for the ISE. These rate effects are characterized in terms of the rate sensitivity of the hardness, , where H is the hardness and is an effective strain rate in the plastic volume beneath the indenter. can be measured using indentation creep, load relaxation, or rate change experiments. The activation volume V∗, calculated based on which can traditionally be used to compare rate sensitivity data from a hardness test to conventional uniaxial testing, is calculated. Using materials with different stacking fault energy and specimens with different levels of work hardening, we demonstrate how increasing the dislocation density affects V∗; these effects may be taken as a kinetic signature of dislocation strengthening mechanisms. We noticed both H and exhibit an ISE. The course of V∗ vs. H as a result of the ISE is consistent with the course of testing specimens with different level of work hardening. This result was observed in both materials. This suggests that a dislocation mechanism is responsible for the ISE. When the results are fitted to a strain gradient plasticity model, the data at deep indents (microhardness and large nanoindentation) exhibit a straight-line behavior closely identical to literature data. However, for shallow indents (nanoindentation data), the slope of the line severely changes, decreasing by a factor of 10, resulting in a “bilinear behavior”.     

Determination of mechanical properties by instrumented indentation

The paper reviews the current state of the depth-sensing indentation (sometimes called nanoindentation), where the information on material behaviour and properties is obtained from the indenter load and depth, measured continuously during loading and unloading. It is shown how the contact parameters and principal characteristics are determined using pointed or spherical indenters. Indentation tests can be used for the measurement of hardness and elastic modulus, and also of the yield stress and for the construction of stress–strain diagrams, for the determination of the work of indentation and its components. Most devices use monotonic loading and unloading, but some also enable measurement under a small harmonic signal added to the basic monotonously increasing load. This makes possible continuous measurement of contact stiffness and the study of dynamic properties and the determination of properties of coatings. One section is devoted to the measurement on viscoelastic-plastic materials, where the delayed deforming must be considered during the measurement as well as in data evaluation. Instrumented indentation can also be used for the study of creep under high temperatures. The paper also discusses the errors arising in depth-sensing measurements and informs briefly about some other possibilities of the method.     

Characterisation of non-linear viscoelastic foods by the indentation technique

A method to determine the non-linear viscoelastic constitutive constants from indentation force–displacement data corresponding to different indentation speeds has been developed. The method consists of two parts. In the first part, the force–displacement data is expressed as two functions which represent the strain and the time-dependent responses, respectively. From these functions, the time-dependent constants and the instantaneous force–displacement response are obtained. In the second part, the strain-dependent variables are determined from the instantaneous force–displacement response through an inverse analysis based on the Levenberg–Marquardt method. The method was verified by numerical experiments using the properties of cheese as examples.     

The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape

The contact of an indenter of arbitrary shape on an elastically anisotropic half space is considered. It is demonstrated in a theorem that the solution of the contact problem is the one that maximizes the load on the indenter for a given indentation depth. The theorem can be used to derive the best approximate solution in the Rayleigh-Ritz sense if the contact area is a priori assumed to have a certain shape. This approach is used to analyze the contact of a sphere and an axisymmetric cone on an anisotropic half space. The contact area is assumed to be elliptical, which is exact for the sphere and an approximation for the cone. It is further shown that the contact area is exactly elliptical even for conical indenters when a limited class of Green's functions is considered. If only the first term of the surface Green's function Fourier expansion is retained in the solution of the axisymmetric contact problem, a simpler solution is obtained, referred to as the equivalent isotropic solution. For most anisotropic materials, the contact stiffness determined using this approach is very close to the value obtained for both conical and spherical indenters by means of the theorem. Therefore, it is suggested that the equivalent isotropic solution provides a quick and efficient estimate for quantities such as the elastic compliance or stiffness of the contact. The “equivalent indentation modulus”, which depends on material and orientation, is computed for sapphire and diamond single crystals.     

Atomistic simulations of elastic deformation and dislocation nucleation during nanoindentation

Nanoindentation experiments have shown that microstructural inhomogeneities across the surface of gold thin films lead to position-dependent nanoindentation behavior [Phys. Rev. B (2002), to be submitted]. The rationale for such behavior was based on the availability of dislocation sources at the grain boundary for initiating plasticity. In order to verify or refute this theory, a computational approach has been pursued. Here, a simulation study of the initial stages of indentation using the embedded atom method (EAM) is presented. First, the principles of the EAM are given, and a comparison is made between atomistic simulations and continuum models for elastic deformation. Then, the mechanism of dislocation nucleation in single crystalline gold is analyzed, and the effects of elastic anisotropy are considered. Finally, a systematic study of the indentation response in the proximity of a high angle, high sigma (low symmetry) grain boundary is presented; indentation behavior is simulated for varying indenter positions relative to the boundary. The results indicate that high angle grain boundaries are a ready source of dislocations in indentation-induced deformation.     

Elevated temperature nanoindentation characterization of poly(para-phenylene vinylene) conjugated polymer films

Temperature dependent mechanical properties of poly(p-phenylene vinylene) (PPV) were investigated using quasi-static (QS) and dynamic nanoindentation (NI) at temperatures over the range of 25 to 100 °C. The reduced modulus decreased from about 4.40 GPa to 3.64 GPa over this temperature range. The plasticity indices at all measurement temperatures were lower than the critical value of 0.875, characterizing material “sink-in”, rather than “pile-up” during measurements. The plasticity index showed a non-monotonic trend, with a minimum value at around 70 °C. Analysis of indentation stress relaxation data, obtained at different temperatures, was also performed using generalized Maxwell viscoelastic models. From these analyses, a relaxation mode, with a characteristic relaxation time of approximately 0.5 s, was evident. The characteristic time remained relatively unchanged over the temperature range of 25 to 100 °C. However, the relaxation modulus associated with this mode showed the expected decrease with increase in temperature.     

Surface versus bulk nucleation of dislocations during contact

The indentation of single crystals by a periodic array of flat rigid contacts is analyzed using discrete dislocation plasticity. Plane strain analyses are carried out with the dislocations all of edge character and modeled as line singularities in a linear elastic solid. The limiting cases of frictionless and perfectly sticking contacts are considered. The effects of contact size, dislocation source density, and dislocation obstacle density and strength on the evolution of the mean indentation pressure are explored, but the main focus is on contrasting the response of crystals having dislocation sources on the surface with that of crystals having dislocation sources in the bulk. When there are only bulk sources, the mean contact pressure for sufficiently large contacts is independent of the friction condition, whereas for sufficiently small contact sizes, there is a significant dependence on the friction condition. When there are only surface dislocation sources the mean contact pressure increases much more rapidly with indentation depth than when bulk sources are present and the mean contact pressure is very sensitive to the strength of the obstacles to dislocation glide. Also, on unloading a layer of tensile residual stress develops when surface dislocation sources dominate.