Axisymmetric Indentation Research Articles

Indentation of Piezoelectric Ceramics: Theory, Experiments and Applications

AbstractA new method is proposed for the characterization of properties of piezoelectric materials using depth-sensing indentation involving both mechanical and electrical measurements. First, a rigorous general theory is presented for axisymmetric indentation of piezoelectric solids with anisotropic properties. The theoretical results facilitate the prediction of the indentation load versus the depth of penetration of indenter into the substrate, as well as some transient electrical effects for different electrical boundary conditions. Used in conjunction with instrumented indentation experiments at the nanoscopic, microscopic or macroscopic size scales, these results lead to the prediction of some of the elastic, dielectric and piezoelectric constants as well as the activation energy for depolarization. The predictions of the theory as well as the validity of the approach have been substantiated further with detailed indentation experiments on PZT-4 and barium titanate using either a conducting indenter or an insulated indenter. The theoretical predictions of the coupled electrical-mechanical indentation of piezoelectric solids have also been checked with finite element analyses. The implications of the proposed method for the design and characterization of piezoelectric materials and for quality control in commercial production are also addressed.

Enhanced consolidation in brittle geomaterials susceptible to damage

This paper examines consolidation behaviour of saturated geomaterials with a matrix component which is susceptible to damage. Finite-element-based computational model accounts for the alteration in both the deformability and permeability characteristics of the porous material due to damage evolution. The isotropic damage criteria governing the evolution of elastic stiffness and hydraulic conductivity parameters are characterized by the dependency of the damage variable on the distortional strain invariant. The computational procedure is utilized to evaluate the extent to which the time-dependent axisymmetric indentation behaviour of a rigid circular punch on a poroelastic half-space can be influenced by the damage evolution in the porous skeleton. © 1998 John Wiley & Sons, Ltd.

Improved theory for contact indentation of sandwich panels

Core crushing and large face-sheet deflections are incorporated in a theory for contact indentation of sandwich panels. The model is based on the assumption of axisymmetric indentation of an infinite elastic face sheet bonded to an elastic-ideally plastic core on a rigid foundation. After core yielding, the problem is separated into an outer region where the core is elastic and an inner region where the core exerts a constant reactive pressure on the face sheet. The outer region is modeled as a plate on an elastic foundation. The inner region is modeled using small-deflection plate theory, first-order large-deflection plate theory, and membrane theory. The unknown plastic radius is found by matching the boundary conditions for the two regions. The extension of the model to orthotropic face sheets is indicated and demonstrated by examples. The predictions are in close agreement with experiments, which contrasts to previously published models where membrane effects and core crushing were neglected. The model can be used for improved modeling of impact on sandwich panels.

Determination of Elastic Moduli of Thin Films Using Inverse Technique. Discussion on Data Sampling in Identation Method.

This paper describes a method for evaluating material properties of multilayered system from data obtained by indentation testing. The measurement information collected from the penetration force-depth curves are employed for identifying elastic moduli of thin films. The axisymmetric indentation problem is first analyzed on the basis of the three-dimensional axisymmetric elasticity theory. Then, an analysis of an elastic contact problem, in which an elastic axisymmetric indenter penetrates an elastic half-space coated with multilayered thin films, is presented. Next, an inverse procedure for determining Young's modulus, Poisson's ratio of thin films and radius of the contact area is described under the condition that the elastic moduli of the substrate and indenter, and thicknesses for layers are known. For an inverse analysis using the complex method, effective sampling of data obtained by penetrating indenters with various radii of curvature into the coated substrate is presented.

Inaccuracies in Sneddon's Solution for Elastic Indentation by a Rigid Cone and their Implications for Nanoindentation Data Analysis

AbstractMethods currently used for analyzing nanoindentation load-displacement data to determine a material's hardness and elastic modulus are based on Sneddon's solution for the indentation of an elastic half-space by a rigid axisymmetric indenter. Although this solution is widely used, no attempts have been made to determine how well it works for conditions of finite deformation, as is the case in most nanoindentation experiments with sharp indenters. Analytical and finite element results are presented which show that corrections to Sneddon's solution are needed if it is to be accurately applied to the case of deformation by a rigid cone. Failure to make the corrections results in an underestimation of the load and contact stiffness and an overestimation of the elastic modulus, with the magnitude of the errors depending on the angle of the indenter and Poisson's ratio of the half-space. For a rigid conical indenter with a half-included tip angle of 70.3°, i.e., the angle giving the same area-to-depth ratio as the Berkovich indenter used commonly in nanoindentation experiments, the underestimation of the load and contact stiffness and overestimation of the elastic modulus may be as large as 13%. It is shown that a simple first order correction can be achieved by redefining the effective angle of the indenter in terms of the elastic constants. Implications for the interpretation of nanoindentation data are discussed.

Finite element analysis of the critical ratio of coating thickness to indentation depth for coating property measurements by nanoindentation

The nanoindentation process of a hard coating on a softer substrate has been simulated by the finite element method (FEM). The frequently used three-face Berkovich indenter was simulated by an axisymmetric conical indenter which has the same projected area-depth function as the Berkovich indenter. The present paper discusses the critical ratios of coating thickness to indentation depth at which the substrate has various degrees of effect (2% and 10%) on the indentation response of the layered systems. The influence of the yield strength ratio of the coating to the substrate and the indenter tip geometry in the critical ratio of thickness to depth has been analysed by the finite element simulation. It was found that increasing both the yield strength ratio and the indenter tip radius increases the critical ratio of thickness to depth. Based on the finite element analysis, an empirical equation has been derived, relating the critical ratio of thickness to depth to the yield strength ratio (from 1.5 to 20) and the indenter tip radius (from 0 to 2.0 μm).

Measuring the elastic properties of anisotropic materials by means of indentation experiments

The unloading process in an indentation experiment is usually modeled by considering the contact of a rigid punch with an elastically isotropic half space. Here we extend the analysis to elastically anisotropic solids. We review some of the basic formulae for describing the indentation of elastically anisotropic solids with axisymmetric indenters. We show how the indentation modulus can be calculated for arbitrary anisotropic solids and give results for solids with cubic crystal symmetry. We have calculated the contact stiffness for a flat triangular punch on a half space for various anisotropic materials. The indentation modulus for a triangular indenter is typically 5–6% higher than that for an axisymmetric indenter and varies only slightly with the orientation of the indenter in the plane of the indentation. We have conducted microindentation experiments to measure the indentation moduli of differently oriented surfaces of both cubic and hexagonal single crystals. For copper and β-brass, the (111) indentation moduli are approximately 10 and 25% larger than the {100} modulus. The (110) moduli are typically slightly smaller than the (111) moduli. The indentation modulus of zinc varies by as much as a factor of two, depending on the sample orientation. The hardnesses of the single crystals do not vary much with the orientation of the plane of indentation. For /gb-brass, the hardness of a {110} surface is only about 13% lower than the hardness of a {100} or {111} surface; for copper, the {110} hardness is 6% higher than for the other orientations. For zinc the maximum change in hardness with orientation is 20%.

On the problem of a detached anchor plate embedded in a crack

The present paper examines the elastostatic problem related to the axisymmetric indentation of the single surface of a penny-shaped crack by a rigid disc anchor. The governing integral equation is solved in a numerical fashion to evaluate the axial stiffness characteristics of the indenting disc anchor and the stress intensity factors at the tip of the penny-shaped crack.

Axisymmetric indentation of a circular stamp with adhesion into an elastic half-space with a spherical cavity

On the basis of the expansion formulas of the vector solutions of the Lame equations in cylindrical and spherical coordinates, the problem of a circular stamp is formulated in the form of an integro-algebraic system of equations. By the method of orthogonal polynomials, it is reduced to a collection of infinite systems of linear algebraic equations, for which the method of reduction is justified. Formulas for the normal and tangential stresses under the stamp are given.

Axisymmetric indentation of an elastic layer supported by a Winkler foundation

Axisymmetric indentation problems of an elastic layer supported by a Winkler foundation are studied in this paper. The indentation on the upper surface is made by a rigid axisymmetric conical, paraboloidal or ellipsoidal indenter. Fundamental solutions for an arbitrary surface load are derived first. The above problems are formulated into an integral equation which is solved numerically. Extensive results are provided for contact radii, displacements and contact pressures. A layer on a rigid smooth base is treated as a special case of the Winkler foundation. The associated analytical solutions for the rigid indenters on a half space arc retrieved as the layer depth approaches infinity. Conversely, useful relations are also derived from thin plate theory for relatively shallow layers.

Finite element analysis of cone indentation

An elastoplastic analysis of axisymmetric conical indentation has been made using the finite element method. Studies of the shapes of plastic zones and the effects of shear modulus, bulk modulus and indenter angle on the hardness of hard and soft materials have been made. The results show that the shapes of the plastic zones depend strongly on both indenter angle and the ratio of Young's modulus to yield strength (E σy). It is also found that the bulk and shear moduli have large effects on the hardness of hard and soft materials, respectively.

Approximate Analysis of Penetration of a Cylindrical Roller Into a Thin Elastic Coating

An approximate analytical solution to the problem of two-dimensional indentation of a frictionless cylinder into a thin elastic coating bonded to a rigid substrate has been obtained using the approach introduced by Matthewson for axisymmetric indentation. We show by comparing the results of the approximate solution to the exact solutions and to finite element calculations that the approximate solution is accurate for a/h> 2. The advantage of this approach is that the results are expressed in closed form and the accuracy of the approximate solution improves with increasing values of a/h. For a/h>2, for a given load, the theory overestimates the value of a/h compared to the exact solution by less than 10 percent. In many experimental studies and in preliminary design, it is convenient to have closed-form solutions exhibiting the dependence of the parameters.

Non-uniqueness and bifurcation in annular and planar channel flows

High-Reynolds-number steady flow in an annular pipe which encounters a shallow axisymmetric expansion or indentation in the walls is studied using interactive boundary-layer theory. The flow upstream of the indentation (x < 0) is fully developed; the ratio of the shear rate on the outer wall to that on the inner wall is denoted by ρ (0 < ρ < 1): similarity solutions are found for the case where the wall perturbations are proportional to . The solution is unique in a constriction, when the pressure gradient (represented by a parameter b) is favourable (b < 0). In an expansion, however, with an adverse pressure gradient, three different solutions are found if b exceeds a critical value bc. When ρ ≠ 1, one of these solutions, representing a flow that is attached on the inner wall and separated (i.e. has negative wall shear) on the outer, is a continuation of the unique doubly attached flow at small b. The other two, one separated on the inner and not the outer wall and the other separated on both walls, arise from a saddle-node bifurcation at b = bc. The doubly separated flow is never stable, as observed in diffusers. In the case of a planar channel (ρ = 1) symmetry is restored, and the non-uniqueness arises through a supercritical pitchfork bifurcation. This agrees with previous computations on channel flow, but not with Jeffery-Hamel flow, for which the bifurcation is subcritical.

Some Axisymmetric Problems for Layered Elastic Media: Part II—Solutions for Annular Indenters and Cracks

In Part I, the multiple contact region solutions for an axisymmetric indenter were presented. The solution technique utilized integral transforms and singular integral equations. The emphasis there was the study of the conditions of contact as a function of the physical parameters of the indenter and the layered elastic half space. The method and results were similar to those for the analogous plane-strain problem that was studied in Shield and Bogy (1989). However, several differences in detail were required for the analysis of the axisymmetric geometry. In this Part II, the solution of Part I is used to study some related problems that have been considered previously in the literature for homogeneous half spaces. First we solve the problem of the axisymmetric annular indenter for the layered half space. Multiple contact region solutions are studied and the problem of an axisymmetric punch with internal pressure is solved for the layered half space and also for the special case of a layer with a traction-free lower surface. Finally, the problem of an annular crack in a homogeneous or layered structure is solved.

Some Axisymmetric Problems for Layered Elastic Media: Part I—Multiple Region Contact Solutions for Simply-Connected Indenters

The contact of a flat, simply-connected axisymmetric indenter with a layered elastic half space is examined. The problem is mathematically formulated using integral transforms to derive singular integral equations for the contact pressure. The solution of these equations is obtained by expansion in orthogonal polynomials. The solution predicts complete contact between the indenter and the surface of the layered half space only for a restricted range of the material and geometrical parameters. Outside of this range, solutions exist with multiple contact regions. A parameter space is divided into zones for single and multiple contact solutions and comparisons are made with the solutions for the analogous plane-strain problem.

Internally indented penny-shaped cracks: A comparison of analytical and boundary integral equation estimates

The problem of the axisymmetric internal indentation of a penny-shaped crack by a rigid circular inclusion is discussed. The paper presents a comparison of analytical and boundary integral equation results for the stress intensity factor at the boundary of the penny-shaped crack indented by a smooth inclusion. Numerical results presented in the paper examines the influence of features such as adhesion at the inclusion-elastic medium interface and finite geometry of the elastic solid containing the penny shaped crack.

Interaction between internal loading and surface indentation for a transversely isotropic elastic half-space

An analytical solution is given for the axisymmetric indentation problem of a half-space loaded by internal forces. Closed form results are presented for an interior point force and a flat-ended indenter. The interaction between internal and external loadings is discussed using numerical results and graphs.

Plastic Indentation of Steel Cylinders With Circular Punches

The detailed deformation characteristics involved in axisymmetric plastic indentation was examined. Circular punches with flat and hemispherical ends were indented into specimens prepared from SAE 1215 steels under as-received and annealed conditions. The load-displacement curves were recorded, and measurements were made of the geometrical changes of the workpieces and of the strain distributions for various punch-workpiece dimensions. Furthermore, the mode of deformation was examined by revealing the flow lines, and fracturing of the workpiece was also observed.

On the Plastic Flow Beneath a Blunt Axisymmetric Indenter

A new approach to large strain plasticity problems in which the material is considered to behave in a plastic-elastic fashion, instead of as a plastic-rigid body, is applied to the axisymmetric blunt indenter. The ratio of the mean stress on the punch face to the uniaxial flow stress of the material (constraint factor C) is found to be 2.82 for an extensive specimen. However, it is shown that a small part of the punch face is elastically loaded, and if the loaded punch area is assumed equal to the size of the plastic impression, then the constraint factor to be used is 3.00 instead of 2.82. This is the value to be used in interpreting the ordinary brinell test. Hardness values are shown to be independent of the degree of friction on the face of a blunt indenter and of the elasticity of the indenter. The amount of material required beneath an axisymmetric indenter in order that there be no upward flow is found to be about 2.6 times the diameter of the impression for steel. However, the exact value depends on Young’s modulus of elasticity, Poisson’s ratio, material hardness, and the depth of the impression relative to the diameter of the indenting sphere. When two opposing axisymmetric indenters are employed, the influence of one on the other will be less than one percent when their spacing is approximately 11.5 times the indentation diameter. A blunt indenter may be defined as one which gives no upward flow in a hardness test. Upward flow should be avoided in hardness testing since it causes the mean stress on the punch face to produce a given impression to be sensitive to friction and the tendency of the metal to strain harden. Upward flow may be prevented by use of an extensive specimen relative to the depth of the impression, large indenter angle (160–180 deg), and high indenter friction (rough surface and no lubricant). The flow stress measured by a blunt indenter is that corresponding to the onset of plastic flow. When upward flow is permitted, the flow stress measured by an indentation hardness test will correspond to an appreciable plastic strain which increases as the included angle of the indenter decreases. The quantity measured by an indenter that performs with upward flow is, therefore, quite ambiguous when the material tested strain hardens.

Elastoplastic analysis of plane-strain and axisymmetric flat punch indentation by the finite-element method

An elastoplastic analysis of plane-strain and axisymmetric flat punch indentation into a specimen of finite dimensions was made by the finite-element method, using an improved finite-element representation. Studies were made on the development of the plastic zone, the load-displacement relationships, and the stress and strain distributions during continued loading, with variations in the punch friction and specimen dimensions. Unloading was considered also and the residual stresses were examined.