Mechanics of indentation for piezoelectric thin films on elastic substrate

Frictionless normal indentation problem of rigid flat-ended cylindrical, conical and spherical indenters on piezoelectric film, which is either in frictionless contact with or perfectly bonded to an elastic half-space (substrate), is investigated. Both conducting and insulating indenters are considered. With Hankel transform, the general solutions of the homogeneous governing equations for the piezoelectric layer and the elastic half-space are presented. Using the boundary conditions for a vertical point force or a point electric charge, and the boundary conditions on the film/substrate interface, the Green’s functions can be obtained by solving sets of simultaneous linear algebraic equations. The solution of the indentation problem is obtained by integrating these Green’s functions over the contact area with unknown surface tractions or electric charge distribution, which will be determined from the boundary conditions on the contact surface between the indenter and the film. The solution is expressed in terms of dual integral equations that are converted to a Fredholm integral equation of the second kind and solved numerically. Numerical examples are also presented. The comparison between two film/substrate bonding conditions is made. It shows that the indentation rigidity of the film/substrate system is lower when the film is in frictionless contact with the substrate. The effects of the Young’s modulus and Poisson’s ratio of the elastic substrate, indenter electrical condition and indenter prescribed electric potential on the indentation responses are presented.     

Indentation responses of piezoelectric films ideally bonded to an elastic substrate

The influences of elastic substrate on the indentation force, contact radius, electric potential and electric charge responses of piezoelectric film/substrate systems are investigated by the integral transform method. The film is assumed to be ideally bonded to the substrate and the contact interaction between the indenter and the film is assumed to be frictionless, with three kinds of axisymmetric insulating and conducting indenters (i.e., punch, cone and sphere) considered. Obtained results show that when the ratio of the contact radius to the film thickness is close to zero, the influences of the elastic substrate disappear and the indentation behaviors converge to the piezoelectric half space solutions while the indentation responses approach the corresponding ones of elastic half space as the ratio gets to infinity. The transition between the piezoelectric and the elastic half space indentation solutions for the film/substrate system is quantified in terms of the film thickness and the elasticity of the substrate. Finite element analysis on an insulating sphere indentation is conducted to verify the numerical calculations and good agreement is observed. The obtained results are believed to be useful for developing experimental techniques to extract the material properties of piezoelectric film/substrate systems.     

Cylindrical nano-indentation on metal film/elastic substrate system with discrete dislocation plasticity analysis: A simple model for nano-indentation size effect

The cylindrical nano-indentation on metal film/elastic substrate is computationally studied using two-dimensional discrete dislocation plasticity combined with the commercial software ANSYS®, with a focus on the storage volume for geometrically necessary dislocations (GNDs) inside the films and the nano-indentation size effect (NISE). Our calculations show that almost all GNDs are stored in a rectangular area determined by the film thickness and the actual contact width. The variations of indentation contact width with indentation depth for various film thicknesses and indenter radii are fitted by an exponential relation, and then the GND density underneath the indenter is estimated. Based on the Taylor dislocation model and Tabor formula, a simple model for the dependence of the nano-indentation hardness of the film/substrate system on the indentation depth, the indenter radius and the film thickness is established, showing a good agreement with the present numerical results.     

Asymptotic modeling of the impact of a spherical indenter on an elastic half-space

The impact of a rigid sphere on a homogeneous, isotropic elastic half-space in the absence of friction and adhesion is considered. The influence of the superseismic stage immediately following the moment of first contact upon the impact process is investigated in the frame of the Hertzian impact theory. The first order asymptotic approximation for the contact force in a three-dimensional dynamic contact problem with the slowly moving contact zone boundary is obtained and the corresponding asymptotic model of impact is developed. The motion of the indenter as it indents and rebounds from the elastic medium is analytically described. Explicit formulas are derived for the peak indentation depth, contact time, and rebound velocity as functions of the initial impact velocity, indenter mass, and characteristics of the elastic half-space.     

Determining the Elastic Modulus of Compliant Thin Films Supported on Substrates from Flat Punch Indentation Measurements

Instrumented indentation is a technique that can be used to measure the elastic properties of soft thin films supported on stiffer substrates, including polymer films, cellulosic sheets, and thin layers of biological materials. When measuring thin film properties using indentation, the effect of the substrate must be considered. Most existing models for determining the properties of thin films from indentation measurements were developed for metal and dielectric films bonded to semiconductor substrates and have been applied to systems with film-substrate modulus ratios between 0.1 and 10. In the present work, flat punch indentation of a thin film either bonded to or in contact with a substrate is examined using finite element modeling. A broad range of film-substrate modulus ratios from 0.0001 to 1 are investigated. As the substrate is effectively rigid compared to the film when the film-substrate modulus ratio is less than 0.0001, the results are also useful for understanding systems with lower film-substrate modulus ratios. The effects of the contact radius, film thickness, elastic properties, and friction between the film and the substrate on the measured stiffness were quantified using finite element modeling in order to understand how the elastic properties of the film can be extracted from indentation measurements. A semi-analytical model was developed to describe the finite element modeling results and facilitate the use of the results to analyze experimental measurements. The model was validated through analysis of indentation measurements of thin polyethylene sheets that were supported on substrates of various stiffness.     

Effect of geometrical and material parameters in nanoindentation of layered materials with an interphase

Indentation models for thin layer-substrate geometry with an interphase have been developed. The interphase can be modeled either as a nonhomogeneous layer or as a homogeneous layer. Between the two models of the interphase, contact depth and critical interfacial stresses are compared to find the effect of indentation area, film and substrate Young’s moduli, and the interphase and film thicknesses. Although contact depth is found not to be sensitive to the type of interphase model used, critical interfacial stresses are significantly different (up to 15%) for film to substrate elastic Young’s moduli ratios of more than 25. A formal sensitivity analysis based on design of experiments shows that on critical interfacial stresses, interphase to film thickness ratio and film to substrate Young’s moduli ratio has the most impact, while type of elastic moduli variation in the interphase and indentor width to film thickness ratio has the least impact.     

A numerical analysis of spherical indentation response of thin hard films on soft substrates

In this paper, finite element simulations of spherical indentation of a thin hard film deposited on a soft substrate are carried out. The primary objective of this work is to understand the operative mode of deformation of the film corresponding to various stages of indentation. The transition from contact dominant behaviour to that governed by flexure of the film on the plastically yielding substrate is investigated from analysis of the load versus displacement curve as well as the stress distribution in the film. It is found that onset of bending deformation in the film occurs when the contact radius is about 0.2–0.3 of the film thickness. Further, distinct membrane stresses arise in the film for indentation depth greater than half the film thickness. The implications of these results on indentation fracture of the film are briefly discussed. Finally, the effects of substrate yield strength and presence of residual stresses on the indentation response are examined.     

二维材料力学行为的压痕测试

准确了解二维材料的力学性能对于推动其应用具有重要意义, 无基底压痕技术是目前最广泛采用的二维材料力学性能测试方法之一, 本文综述了二维材料压痕研究的最新进展以及所面临的问题, 并对将来的研究工作进行了展望.无基底压痕技术是将二维材料转移到带有沟槽或柱形孔的基底上, 制备二维材料"梁"和"鼓"模型, 然后利用原子力显微镜测量其在压针作用下的载荷--位移关系, 最后通过基于连续介质薄膜导出的压痕响应分析模型拟合实验结果, 估算出二维材料的弹性模量和本征强度.由于二维材料的厚度远小于连续介质薄膜, 来自于压头以及基底孔侧壁的范德华力对二维材料的压痕响应具有显著影响, 造成二维材料与传统压痕分析模型中的基本假设不符, 导致不能准确预测二维材料的弹性模量; 另外, 由于传统压痕模型无法准确描述二维材料在大变形下的非线性行为, 以及由缺陷等引起的应力集中, 导致由压痕测试表征的二维材料(特别是多晶二维材料)本征强度具有较大的偏差. 因此, 一方面需要正确了解由压痕技术获得的二维材料力学性能, 另一方面还需对目前的研究方法做进一步的改进和完善.      

Indentation tests investigation of mechanical behavior of two-dimensional materials

准确了解二维材料的力学性能对于推动其应用具有重要意义, 无基底压痕技术是目前最广泛采用的二维材料力学性能测试方法之一, 本文综述了二维材料压痕研究的最新进展以及所面临的问题, 并对将来的研究工作进行了展望.无基底压痕技术是将二维材料转移到带有沟槽或柱形孔的基底上, 制备二维材料"梁"和"鼓"模型, 然后利用原子力显微镜测量其在压针作用下的载荷--位移关系, 最后通过基于连续介质薄膜导出的压痕响应分析模型拟合实验结果, 估算出二维材料的弹性模量和本征强度.由于二维材料的厚度远小于连续介质薄膜, 来自于压头以及基底孔侧壁的范德华力对二维材料的压痕响应具有显著影响, 造成二维材料与传统压痕分析模型中的基本假设不符, 导致不能准确预测二维材料的弹性模量; 另外, 由于传统压痕模型无法准确描述二维材料在大变形下的非线性行为, 以及由缺陷等引起的应力集中, 导致由压痕测试表征的二维材料(特别是多晶二维材料)本征强度具有较大的偏差. 因此, 一方面需要正确了解由压痕技术获得的二维材料力学性能, 另一方面还需对目前的研究方法做进一步的改进和完善.     

Effect of friction on subsurface stresses in sliding line contact of multilayered elastic solids

A numerical integral scheme based on Fourier transformation approach is employed to investigate the effect of friction on subsurface stresses arising from the two-dimensional sliding contact of two multilayered elastic solids. The analysis incorporates bonded and unbonded interface boundary conditions between the coating layers. Two line contact problems are presented. The first one is the contact problem between a rigid cylinder and a two-layer half space and the second one is the indentation of a multilayered elastic half-space by a flat rigid punch. The effects of the surface coating on the contact pressure distribution and subsurface stress field are presented and discussed.     

Delamination of an elastic film from an elastic–plastic substrate during adhesive contact loading and unloading

Adhesive contact between a rigid sphere and an elastic film on an elastic–perfectly plastic substrate was examined in the context of finite element simulation results. Surface adhesion was modeled by nonlinear springs obeying a force-displacement relationship governed by the Lennard–Jones potential. A bilinear cohesive zone law with prescribed cohesive strength and work of adhesion was used to simulate crack initiation and growth at the film/substrate interface. It is shown that the unloading response consists of five sequential stages: elastic recovery, interface damage (crack) initiation, damage evolution (delamination), film elastic bending, and abrupt surface separation (jump-out), with plastic deformation in the substrate occurring only during damage initiation. Substrate plasticity produces partial closure of the cohesive zone upon full unloading (jump-out), residual tensile stresses at the front of the crack tip, and irreversible downward bending of the elastic film. Finite element simulations illustrate the effects of minimum surface separation (i.e., maximum compressive surface force), work of adhesion and cohesive strength of the film/substrate interface, substrate yield strength, and initial crack size on the evolution of the surface force, residual deflection of the elastic film, film-substrate separation (debonding), crack-tip opening displacement, and contact instabilities (jump-in and jump-out) during a full load–unload cycle. The results of this study provide insight into the interdependence of contact instabilities and interfacial damage (cracking) encountered in layered media during adhesive contact loading and unloading.     

Modeling the influence of the coating deposition technology on the contact interaction characteristics

For a multilayer elastic half-space, we consider an axisymmetric loading model taking into account damage on the interface between the layers. The influence of intermediate layers arising in various coating technologies on the contact and internal stresses occurring in the coating and the substrate under elastic indentation conditions is studied for relatively rigid and nonrigid coatings.     

Analysis of elastic and plastic deformation associated with indentation testing of thin films on substrates

A study has been made of the elastic and plastic deformation associated with submicrometer indentation of thin films on substrates using the finite element method. The effects of the elastic and plastic properties of both the film and substrate on the hardness of the film/substrate composite are studied by determining the average pressure under the indenter as a function of the indentation depth. Calculations have been made for film/substrate combinations for which the substrate is either harder or softer than the film and for combinations for which the substrate is either stiffer or more compliant than the film. It is found, as expected, that the hardness increases with indentation depth when either the yield strength or the elastic modulus of the substrate is higher than that of the film. Correspondingly, the hardness decreases with indentation depth when the yield strength or elastic modulus of the substrate is lower than that of the film. Functional equations have been developed to predict the hardness variation with depth under these different conditions. Finite element simulation of the unloading portion of the load displacement curve permits a determination of the elastic compliance of the film/substrate composite as a function of indentation depth. The elastic properties of the film can be separated from those of the substrate using this information. The results are in good agreement with King's analytical treatment of this problem.     

Delamination mechanism maps for a strong elastic coating on an elastic–plastic substrate subjected to contact loading

Hard wear resistant coatings that are subjected to contact loading sometimes fail because the coating delaminates from the substrate. In this report, systematic finite element computations are used to model coating delamination under contact loading. The coating and substrate are idealized as elastic and elastic–plastic solids, respectively. The interface between coating and substrate is represented using a cohesive zone law, which can be characterized by its strength and fracture toughness. The system is loaded by an axisymmetric, frictionless spherical indenter. We observe two failure modes: shear cracks may nucleate just outside the contact area if the indentation depth or load exceeds a critical value; in addition, tensile cracks may nucleate at the center of the contact when the indenter is subsequently removed from the surface. Delamination mechanism maps are constructed which show the critical indentation depth and force required to initiate both shear and tensile cracks, as functions of relevant material properties. The fictitious viscosity technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces allows us to explore a wider parametric space that a conventional cohesive model cannot handle. Numerical results have also been compared to analytical analyses of asymptotic limits using plate bending and membrane stretching theories, thus providing guidelines for interpreting the simulation results.     

Indentation of a round punch into a rough elastic half-space

We consider the axisymmetric contact problem on the indentation of a round punch with plane base into an elastic half-space. Roughness is taken into account in the framework of the well-known model in which the local surface displacements are proportional to some power of the contact pressures at the same point. To study the problem, we use the theory of nonlinear integral equations of Hammerstein type together with the mathematical technique of orthogonal Legendre polynomials and the contraction mapping principle. We obtain an approximate analytic solution of the problem, numerically analyze the results, and reveal typical laws of variation of the main mechanical variables.     

Adhesion-induced instabilities in elastic and elastic-plastic contacts during single and repetitive normal loading

Adhesive interaction in spherical contacts was modeled with the Lennard-Jones (L-J) potential. Elastic adhesive contact was analyzed by the equivalent system of a rigid sphere with reduced radius of curvature and a half-space of effective elastic modulus. The critical gap at the instant of abrupt surface contact (jump-in) and separation (jump-out) was determined from the deformed surface profile of the elastic half-space and geometrical relationships. A finite element model of a rigid sphere and an elastic-plastic half-space was used to examine elastic-plastic adhesive contact. Surface adhesion was modeled by nonlinear springs with a force-displacement relationship governed by the L-J potential. The evolution of the interfacial force and the central gap distance as well as the occurrence of jump-in and jump-out instabilities were investigated in terms of the Tabor parameter, plasticity parameter, and dimensionless maximum normal displacement. The force-displacement response due to several approach-retraction cycles was interpreted in the context of elastic and plastic shakedown behaviors using dimensionless parameters.     

Stress disturbances caused by the inhomogeneity in an elastic half-space subjected to contact loading

Inhomogeneities can increase localized stress and cause microstructural alterations to initiate fatigue failures in rolling elements under cyclic contact loading. To study the stress disturbances created by the inhomogeneity, a two-dimensional contact stress analysis is presented for a cylindrical indenter sliding on an elastic half-space containing an inhomogeneity of arbitrary shape. The boundary element method is used to analyze the contact problem, where actual contact boundary, contact pressure as well as tractions and displacements at inhomogeneity–substrate interface are determined by solving a set of integral equations numerically. Numerical results are presented to investigate effects and the stress disturbances caused by the inhomogeneity with various locations, sizes and material properties of inhomogeneity. The results also show that hard inclusions are more detrimental than soft deformable particles in rolling contact elements.     

High speed indentation of an elastic half-space by conical or wedge-shaped indentors

Self-similar problems of indentation of an elastic half-space by rigid cones or wedges are solved, assuming perfect adhesion, when the velocity of indentation is large enough for the area of contact to spread faster than the speed of P-waves. In contrast to the earlier study of the wholly subsonic case [2], the present problems can be solved in closed form without approximation. It emerges, too, that the no slip condition would be satisfied for a range of values of a finite coefficient of friction, in contrast to the situation in [2], where any finite friction is bound to allow some slip. A variety of wave fronts exist in the present problems and all of their amplitudes are found explicitly and discussed.School of Mathematics, University of Bath     

A numerical approach to indentation technique to evaluate material properties of film-on-substrate systems

Spherical indentation approach (Lee et al., 2005, Lee et al., 2010) for the evaluation of bulk material properties is extended to that for elastic–plastic properties of film-on-substrate systems. Our interest focuses on single isotropic, metallic, and elastic–plastic film on a substrate, and we do not consider the size effects in plasticity behavior. We first determine the optimal data acquisition location, where the strain gradient is the least and the effect of friction is negligible. Dimensional analysis affords the mapping parameters as functions of normalized indentation variables. An efficient way is further introduced to reduce both the number of analyses and the regression order of mapping functions. The new numerical approach to the film indentation technique is then proposed by examining the finite element solutions at the optimal point. With the new approach, the values of elastic modulus, yield strength, and strain-hardening exponent of film materials are successfully obtained from the spherical indentation tests. We have shown that the effective property ranges such as indenter properties, substrate modulus, and E/E_s ratio can be extended without additional simulations and even loss of accuracy. For other ranges of variables or other properties, which are not dealt with in this study, this methodology is applicable through resetting FEA variables and finding proper normalized parameters.     

Influence of penetration depth and mechanical properties on contact radius determination for spherical indentation

Knowledge of the relationship between the penetration depth and the contact radius is required in order to determine the mechanical properties of a material starting from an instrumented indentation test. The aim of this work is to propose a new penetration depth–contact radius relationship valid for most metals which are deformed plastically by parabolic and spherical indenters. Numerical simulation results of the indentation of an elastic–plastic half-space by a frictionless rigid paraboloïd of revolution show that the contact radius–indentation depth relationship can be represented by a power law, which depends on the reduced Young’s modulus of the contact, on the strain hardening exponent and on the yield stress of the indented material. In order to use the proposed formulation for experimental spherical indentations, adaptation of the model is performed in the case of a rigid spherical indenter. Compared to the previous formulations, the model proposed in the present study for spherical indentation has the advantage of being accurate in the plastic regime for a large range of contact radii and for materials of well-developed yield stress. Lastly, a simple criterion, depending on the material mechanical properties, is proposed in order to know when piling-up appears for the spherical indentation.