Flat-ended Cylindrical Indenter Research Articles

Novel method for measuring surface residual stress using flat-ended cylindrical indentation

Instrumented indentation is a promising technique for estimating surface residual stresses and mechanical properties in engineering components. The relative difference between the indentation loads for unstressed and stressed specimens was selected as the key parameter for measuring surface residual stresses in flat-ended cylindrical indentations. Based on the equivalent material method and finite element simulations, a dimensionless mapping model with six constants was established between the relative load difference, constitutive model parameters, and normalized residual stress. A novel method for measuring the surface residual stress and constitutive model parameters of metallic material through flat-ended cylindrical indentations was proposed using this model and a mechanical properties determination method. Numerical simulations were conducted using numerous elastoplastic materials with different residual stresses to verify the proposed model; good agreements were observed between the predicted residual stresses and those previously applied in finite element analysis. Flat-ended cylindrical indentation tests were performed on four metallic materials using cruciform specimens subjected to various equibiaxial stresses. The results exhibited good conformance between the stress–strain curves obtained using the proposed method and those from traditional tensile tests, and the absolute differences between the predicted residual stresses and applied stresses were within 40 MPa in most cases.

Indentation behavior of a semi-infinite piezoelectric semiconductor under a rigid flat-ended cylindrical indenter

Indentation behavior of a semi-infinite piezoelectric semiconductor under a rigid flat-ended cylindrical indenter

The revisited phase-field approach to brittle fracture: application to indentation and notch problems

In a recent contribution, Kumar et al. (J Mech Phys Solids 142:104027, 2020) have introduced a comprehensive macroscopic phase-field theory for the nucleation and propagation of fracture in linear elastic brittle materials under arbitrary quasistatic loading conditions. The theory can be viewed as a natural generalization of the phase-field approximation of the variational theory of brittle fracture of Francfort and Marigo (J Mech Phys Solids 46:1319–1342, 1998) to account for the material strength at large. This is accomplished by the addition of an external driving force—which physically represents the macroscopic manifestation of the presence of inherent microscopic defects in the material—in the equation governing the evolution of the phase field. The main purpose of this paper is to continue providing validation results for the theory by confronting its predictions with direct measurements from three representative types of experimentally common yet technically challenging problems: (i) the indentation of glass plates with flat-ended cylindrical indenters and the three-point bending of (ii) U-notched and (iii) V-notched PMMA beams.

Effects of surface stress on the indentation response of an elastic half-space

Nanoindentation technique, which is based on Hertzian contact theory and Sneddon's solutions for the contact between a rigid indenter and an elastic half-space, has been widely investigated due to its practical importance in localized mechanical test of submicron structures. However, both the Hertzian contact theory and Sneddon's solutions do not take into account the contribution of surface stress to contact deformation. In this work, we study the effect of surface stress without the out-of-plane term on the indentation deformation of an elastic half-space by rigid, axisymmetric indenters, including flat-ended cylindrical, conical and spherical indenters. In contrast to classical theories, which are based on physically admissible condition of finite normal stress at contact edge, an alternative condition of geometrical continuity at contact edge is used to determine the contact radius. The numerical results reveal the combinational effects of the surface stress and Poisson's ratio on the load-displacement relationship for the indentation of the elastic half-space. The surface stress causes significant change in shear stress and modest variation in normal stress in the direction normal to the surface of the elastic half-space. The numerical method used in this work offers a feasible approach to study the effect of surface stress on the contact deformation of elastic substrates.

Force relaxation of contact between a flat-ended cylindrical indenter and a poroviscoelastic layer

Interpreting the contact force relaxation between an indenter and a poroviscoelastic material at a fixed displacement to obtain the materials properties is a challenging task, because both fluid drainage and solid viscoelasticity contribute to the relaxation. In this research, the indentation of a poroviscoelastic layer caused by a flat-ended cylindrical indenter is formulated analytically by the displacement functions in the Laplace-transformed domain. The contact force of the indenter at a fixed displacement was expressed by a closed-form in the transformed domain and numerically inverse-transformed to the time domain. The force relaxation is analyzed for combinations of three drainage conditions prescribed on the layer top and two boundary conditions prescribed on the layer bottom. The closed-form solution can approach that of the flat-indentation on a poroviscoelastic half-space, when the layer is very thick compared to the indenter radius. In addition, finite element simulations of the flat-indentation on poroviscoelastic layers were carried out as a comparison for the semi-analytical results. The relaxation ratios at the limits of the drained-nonviscoelastic state, the nondrained-viscoelastic state and the drained-viscoelastic state are shown for different ratios of the layer thickness to the indenter radius, and verified with the corresponding elastic solutions. The effects of the viscoelastic characteristic time on the force relaxations are also presented for different viscoelastic properties and boundary conditions.

Contact Nonlinearity in Indenter–Foam Dampers

AbstractIn this paper, the nonlinear response of indenter–foam dampers is characterized. Those dampers consist of indenters pressed on open-cell foams swollen with wetting liquids. Recently, the authors identified the dominant mechanism of damping in those dampers as poro-viscoelastic (PVE) relaxations as in articular cartilage, one of nature’s best solutions to vibration attenuation. Those previous works by the authors included dynamic mechanical analyses of the indenter–foam dampers under small vibrations, i.e., linear regime. The current study features the dynamic response of similar dampers under larger strains to investigate the nonlinear regime. In particular, the indenter–foam dampers tested in this paper consist of an open-cell polyurethane foam swollen with castor oil. Harmonic displacements are applied on the swollen and pre-compressed foam using a flat-ended cylindrical indenter. Measured forces and corresponding hysteresis (force–displacement) loops are then analyzed to quantify damping performance (via specific damping capacity) and nonlinearities (via harmonic ratio). The effects of strain and strain rates on the damping capacity and harmonic ratio are investigated experimentally. The dominant source of the nonlinearity is identified as peeling at the indenter–foam interface (and quantified via peeling index). A representative model consisting of a linear viscoelastic foam and rate-dependent adhesive interface (slider element with limiting adhesive strength) explains the observed trends in peeling and thus nonlinear dynamic response. Possible remedies to suppress those nonlinearities in future designs of indenter–foam dampers are also discussed.

Axisymmetric Problem for Indentation of Multi-Layer Poroelastic Body

We considered the axisymmetric indentation of poroelastic multi-coating bonded to poroelastic substrate indented by a rigid flat-ended cylindrical indenter. The approach that expressed poroelastic solutions in term of two displacement functions and the transfer matrix method which can formulate the effect of multi-layer composite were applied in this study. Furthermore, we adopted the analytic method that using Laplace-Hankel transform and expressing the normal contact stress at the surface layer as an appropriate series with a Chebyshev orthogonal polynomial to reduce dual integral equations to an infinite system of simultaneous equations in Laplace transform domain. The numerical results of time-dependent axial displacement of rigid cylindrical indenter were presented by performing numerical inverse Laplace transform. The results of axial displacement for one-layer bonded to rigid impervious substrate given by this study were good agreement with previously reported results.

Influence of frictional contact on indentation of elastic layer under surface energy effects

The paper examines the influence of frictional contact on an indentation problem of a finite-thickness elastic layer under a rigid flat-ended cylindrical indenter considering surface energy effects based on Coulomb friction model. The continuum-based approach corresponding to the complete version of Gurtin-Murdoch surface elasticity model is adopted to demonstrate the influence from surface energy. Normal and radial displacement boundary conditions in the contact region corresponding to the frictional indentation problem is formulated in the term of integral equations by employing the relevant displacement Green's function obtained by the use of Hankel integral transform. The contact tractions are solved by applying a collocation technique together with the assumption of a piecewise distribution for the tractions within each discretized annular ring element. The accuracy of proposed solution scheme is validated with existing solutions from classical continuum mechanics for frictional indentation problem of an elastic half-space. Selected numerical results are presented to demonstrate the influence of friction effects on the elastic fields of the layer under the influence of surface energy. It is found that the consideration of frictional contact makes the layer stiffer when compared to the frictionless indentation, and the influence of friction effect is smaller when the surface energy effects are accounted for. In addition, the elastic fields of the layer are clearly size-dependent and the elastic layer becomes stiffer with the presence of surface stresses.

Analytical solution of axisymmetric indentation of multi-layer coating on elastic substrate body

We consider the axisymmetric contact problem of a multi-elastic layer with various elastic constants bonded to an elastic semi-infinite substrate indented by rigid flat-ended cylindrical and spherical indenters. The transfer matrix method is applied to each elastic layer, and dual integral equations are reduced to an infinite system of simultaneous equations by expressing the normal contact stress at the surface elastic layer as an appropriate series with Chebyshev orthogonal polynomials. Numerical results demonstrate the effects of the elastic constant of each elastic layer and the semi-infinite elastic substrate on the radial distribution of the normal contact stress and normal displacement of the free surface of the elastic layer, stress singularity factor at the edge of the cylindrical indenter, and axial load of a rigid indenter which penetrates the multi-layer material to a constant depth. The results of axial load are in good agreement with previously reported results. The numerical results are given for several combinations of the shear modulus of each elastic layer and the substrate. These results will contribute to the establishment of indentation tests for composite materials and serve as guidelines for the design of appropriate mechanical properties of layered materials.

Estimation of Fracture Toughness Using Flat-Ended Cylindrical Indentation

A method is proposed to predict the fracture toughness of in-service structures using the instrumented indentation test. While previous studies have attempted to predict fracture toughness using spherical indenters, we propose the method to predict fracture toughness using flat-ended cylindrical indenters. Using the geometric similarity of a cylindrical indentation test and the Cracked Round Bar (CRB) fracture toughness test, fracture toughness values were derived from a single indentation test by assuming that the load–depth curve of the indentation test is the same as the load–displacement curve of the CRB fracture toughness test. To determine the crack initiation point, the concept of limit load in CRB testing is adopted, and a new load–depth curve is obtained using a suggestion in the standard of fracture toughness test. In order to apply the proposed method directly to in-service structures, the model uses mechanical parameters that can be obtained by indentation testing. The model was verified on metallic materials primarily used in nuclear power plants.

Indentation of a prestretched strain-stiffening elastomer

Strain-stiffening behavior of materials such as rubberlike materials and biological soft tissues is an important phenomenon. In this paper, we proposed a surface Green’s function tensor to describe the strain-stiffening behavior of the stretchable elastomer based on the Gent constitutive model. The surface Green’s function tensor of the Gent constitutive model can be recovered to that of neo-Hookean model, and applied to the indentation problem with a flat-ended cylindrical indenter. The relation between the indentation force and strain-stiffening parameter is analytically derived for equi-biaxial prestretched elastomers. The study shows that the strain-stiffening of the elastomer has a great impact on indentation behaviors, especially for the cases of large prestretches. For a given indentation depth, the indentation force decreases with the increase of the strain-stiffening parameter. For a given stiffening parameter, the indentation force increases with the increase of the prestretches. The proposed surface Green’s function tensor has also potential applications in other fields, such as wetting, cell migration, self-assembly on strain-stiffening materials, etc.

Evaluation of tensile yield strength of high-density polyethylene in flat-ended cylindrical indentation: An analytic approach based on the expanding cavity model

Abstract

Effect of Stress-Fiber Inclusion on the Local Stiffness of Cell Cytoskeleton Probed by AFM Indentation: Insights from a Discrete Network Model

In this paper, we analyze the effect of stress-fiber inclusion on the stiffness of an actin random network. To do this, use a discrete random network model to analyze the elastic response of this system in terms of apparent Young’s modulus. First, we showed that for a flat-ended cylindrical AFM indenter the total indentation force has a linear relation with the indentation depth and the indenter radius in a fibrous network. Using this relation, we concluded that the stiffening effect of the stress-fiber on the fibrous network has a range of effectivity and surprisingly, the stiffening is not maximum when the stress-fiber is immediately under the indenter but, when has a certain distance with it. In addition, when the stress-fiber axis has a specific distance from the loading region, it has negligible effect on the local stiffness of the network. These results shed light on some aspects of the widely used AFM stiffness measurements of cells.

Contact stiffness indentation tomography: Moduli-perturbation approach

The problem of parameter identification for a slightly perturbed inhomogeneous isotropic elastic semi-infinite medium is considered with application to indentation stiffness tomography. A moduli-perturbation approach is applied to evaluate the contact stiffness of the inhomogeneous medium for a frictionless flat-ended cylindrical indenter. The cases of homogeneous spherical, hemispherical, and semi-infinite cylindrical inclusions embedded in a reference homogeneous medium represented by an elastic half-space are considered in detail. The so-called inhomogeneity sensitivity factors are introduced to estimate the characteristic size of inhomogeneity from the indentation data obtained from indenters of three different radii. The effect of elastic invisibility of inclusion to the indentation sensing is discussed. Based on the grid-indentation technique, an approach is proposed for locating and identifying a spherical inhomogeneity buried in an elastic half-space, which utilizes the indentation stiffness data collected from the half-space surface.

Indentation on a half-infinite one-dimensional hexagonal quasi-crystal space by a rigid flat-ended cylindrical indenter with uniform heat flux or temperature

The present paper analytically investigates the contact problem of a half-infinite one-dimensional hexagonal quasi-crystal medium punched by a rigid flat-ended cylindrical indenter with uniform heat flux or temperature. The contact between the two objects is assumed to be frictionless. Based on the general solution, the three-dimensional thermo-elastic coupling field variables in the half-space are explicitly obtained by the generalized potential theory method. For the case of heat flux load, the thermo-elastic field variables are expressed in terms of elementary functions and in closed-forms. In contrast, for the case of uniform temperature load, the corresponding variables are in terms of linear integrals, all the integrands are elementary functions. Furthermore, the maximum shear stress and the von Mises stress in the half-space are also obtained. Some significant physical quantities on the contact plane, such as vertical and radial displacements, normal stresses, temperature and heat flux, are presented as well. An illustrative numerical calculation is performed to show the distributions of the thermo-elastic fields in the vicinity of the contact area. The present solution could be served as the theoretical basis for scanning probe microscopy technology to explore the material properties of one-dimensional hexagonal quasi-crystals.

Surface effect in axisymmetric Hertzian contact problems

Surface effect in three different axisymmetric Hertzian contact models is investigated in this paper with a recently developed elastic theory for nanostructured materials, including a Boussinesq problem, contact problem between a rigid flat-ended cylindrical indenter and an elastic half space as well as contact problem between a rigid spherical indenter and an elastic half space. With the help of the Love's strain function method and Hankel integral transformation, closed-form solutions of the stress and displacement fields at the surface of an elastic half space subjected to a concentrated force are achieved, based on which the interface tractions and displacements in the three different axisymmetric contact problems can be further obtained. It is found that surface effect in these contact problems can be characterized only by an intrinsic length, i.e., the ratio of the bulk surface energy density to the bulk shear modulus of the indented material. When the contact radius is comparable with the intrinsic length, surface effect is much obvious, leading to a serious deviation between the two solutions predicted respectively by the theoretical model developed for nanomaterials and the classical contact model. A more interesting phenomenon is about surface effect on the indentation hardness, which is found to increase with the reduction of the indenter radius when the external load is fixed, or to increase with the decrease of the external load when the indenter radius keeps unchanged. All the results in this paper should be helpful not only for deep understanding of the surface effect on nano-contact behaviors but also for further revealing the nature of surface effect of nano-indentation hardness.

Contact Properties and Adhesion of Incompressible Power-Law Gradient Media with High Gradients

We discuss contact stiffness and adhesion of flat-ended cylindrical indenters with a graded material the elastic coefficient of which is a power-function of the depth with an exponent 1 1 the adhesive properties of contacts change qualitatively from "brittle" to very tough even in the case of a purely elastic material.

Indentation of a Transversely Isotropic Thermoporoelastic Half-Space by a Rigid Circular Cylindrical Punch

Exact solutions to the three-dimensional (3D) contact problem of a rigid flat-ended circular cylindrical indenter punching onto a transversely isotropic thermoporoelastic half-space are presented. The couplings among the elastic, hydrostatic, and thermal fields are considered, and two different sets of boundary conditions are formulated for two different cases. We use a concise general solution to represent all the field variables in terms of potential functions and transform the original problem to the one that is mathematically expressed by integral (or integro-differential) equations. The potential theory method is extended and applied to exactly solve these integral equations. As a consequence, all the physical quantities of the coupling fields are derived analytically. To validate the analytical solutions, we also simulate the contact behavior by using the finite element method (FEM). An excellent agreement between the analytical predictions and the numerical simulations is obtained. Further attention is also paid to the discussion on the obtained results. The present solutions can be used as a theoretical reference when practically applying microscale image formation techniques such as thermal scanning probe microscopy (SPM) and electrochemical strain microscopy (ESM).

Indentation of a stretched elastomer

Indentation has been intensively used to characterize mechanical properties of soft materials such as elastomers, gels, and soft biological tissues. In most indentation measurements, residual stress or stretch which can be commonly found in soft materials is ignored. In this article, we aim to quantitatively understand the effects of prestretches of an elastomer on its indentation measurement. Based on surface Green's function, we analytically derive the relationship between indentation force and indentation depth for a prestretched Neo-Hookean solid with a flat-ended cylindrical indenter as well as a spherical indenter. In addition, for a non-equal biaxially stretched elastomer, we obtain the equation determining the eccentricity of the elliptical contacting area between a spherical indenter and the elastomer. Our results clearly demonstrate that the effects of prestretches of an elastomer on its indentation measurement can be significant. To validate our analytical results, we further conduct correspondent finite element simulations of indentation of prestretched elastomers. The numerical results agree well with our analytical predictions.

Mechanical characterization of nanoclay-filled PDMS thin films

Polydimethylsiloxane (PDMS) reinforced by nanoclay platelets exhibits excellent mechanical properties for microfluidic systems. We tested PDMS-clay nanocomposite thin films under flat-ended cylindrical indentation and measured the elastic modulus and shear strength. A simple formulation based on a shear-lag model, fed by numerical simulations, was introduced to estimate the interfacial shear strength of thin films. Increase in the nanoclay content improved the elasticity of PDMS-clay thin films but reduced their interfacial shear strength. Shear thinning behavior of nanoclay platelets probably reduced the strength of PDMS nanocomposites. The proposed approach can be used for characterization of any polymeric thin films.