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.     

Indentation of a hard film on a soft substrate: Strain gradient hardening effects

The strain gradient work hardening is important in micro-indentation of bulk metals and thin metallic films, though the indentation of thin films may display very different behavior from that of bulk metals. We use the conventional theory of mechanism-based strain gradient plasticity (CMSG) to study the indentation of a hard tungsten film on soft aluminum substrate, and find good agreement with experiments. The effect of friction stress (intrinsic lattice resistance), which is important in body-center-cubic tungsten, is accounted for. We also extend CMSG to a finite deformation theory since the indentation depth in experiments can be as large as the film thickness. Contrary to indentation of bulk metals or soft metallic films on hard substrate, the micro-indentation hardness of a hard tungsten film on soft aluminum substrate decreases monotonically with the increasing depth of indentation, and it never approaches a constant (macroscopic hardness). It is also shown that the strain gradient effect in the soft aluminum substrate is insignificant, but that in the hard tungsten thin film is important in shallow indentation. The strain gradient effect in tungsten, however, disappears rapidly as the indentation depth increases because the intrinsic material length in tungsten is rather small.     

纳米压痕法测磁控溅射铝薄膜屈服应力

为了在考虑残余应力下测量出磁控溅射铝薄膜的屈服应力,提出了一种实验测量方法,通过曲率测试法和球形压头纳米压痕法测出磁控溅射铝薄膜的屈服应力.建立球形压痕力学模型,并用ANSYS对球形压痕进行力学有限元仿真,利用直流磁控溅射技术在硅基上淀积一层l μm厚的铝薄膜,首先通过曲率测试法测量膜内等双轴残余应力,再利用最小二乘曲线拟合法从薄膜/基底系统的球形压头纳米压痕实验数据中提取出铝薄膜的屈服应力,测得磁控溅射铝薄膜的屈服应力为371±89 Mpa.该方法也可以用来研究其他材料的薄膜和小体积材料的力学特性.     

On the Measurement of two Independent Viscoelastic Functions with Instrumented Indentation Tests

In the present paper, a methodology for complete characterization of linear isotropic viscoelastic material with spherical instrumented indentation test is proposed. The developed method allows for measuring two independent viscoelastic functions, shear relaxation modulus and time-dependent Poisson’s ratio, from the indentation test data obtained at non-decreasing loading, but otherwise arbitrary. Finite element modelling (FEM) is relied upon for validating the proposed methodology and for quantifying the influence of experimental variables on the measurements accuracy. Spherical indentation experiments are performed on several viscoelastic materials: polyoxymethylene, bitumen and bitumen-filler mastics. The viscoelastic material functions obtained with the indentation tests are compared with the corresponding results from the standard mechanical tests. Numerical and experimental results presented indicate that the methodology proposed allows mitigating the machine compliance and loading rate effects on the accuracy of the viscoelastic indentation tests.     

An analytical solution for a linear viscoelastic layer loaded with a cylindrical punch: Evaluation of the rebound indentation test with application for assessing viability of articular cartilage

A general closed-form solution for the so-called rebound indentation test’ is obtained for a cylindrical flat-ended punch indenting a linear viscoelastic layer lying on a rigid substrate. Under the assumption of time-independent Poisson's ratio, we derive closed-form analytical expressions for the contact force (in a displacement controlled regime) and for the indentation displacement (in a load-controlled regime) and we consider in detail the case of standard viscoelastic solid. Our results indicate that the rebound displacement (in other words the indentation displacement in the load-controlled stage) is independent of the relaxed elastic modulus and Poisson's ratio, and also of the layer's thickness. Our analytical solution can be used for layered samples of arbitrary materials exhibiting viscoelastic properties; however, since the rebound indentation test has been recently suggested for assessing the viability of biomedical materials, we have applied our theoretical framework to the identification of materials parameters from experiments on articular cartilage. In this context, we have found a pretty good agreement for the rebound deformation, even until the strain becomes relatively large.     

An analytical solution for a linear viscoelastic layer loaded with a cylindrical punch: Evaluation of the rebound indentation test with application for assessing viability of articular cartilage

A general closed-form solution for the so-called rebound indentation test’ is obtained for a cylindrical flat-ended punch indenting a linear viscoelastic layer lying on a rigid substrate. Under the assumption of time-independent Poisson's ratio, we derive closed-form analytical expressions for the contact force (in a displacement controlled regime) and for the indentation displacement (in a load-controlled regime) and we consider in detail the case of standard viscoelastic solid. Our results indicate that the rebound displacement (in other words the indentation displacement in the load-controlled stage) is independent of the relaxed elastic modulus and Poisson's ratio, and also of the layer's thickness. Our analytical solution can be used for layered samples of arbitrary materials exhibiting viscoelastic properties; however, since the rebound indentation test has been recently suggested for assessing the viability of biomedical materials, we have applied our theoretical framework to the identification of materials parameters from experiments on articular cartilage. In this context, we have found a pretty good agreement for the rebound deformation, even until the strain becomes relatively large.     

Effects of large deformation and material nonlinearity on spherical indentation of hyperelastic soft materials

Up to now, the indentation of hyperelastic soft materials has not been completely understood. In this paper, the spherical indentation on hyperelastic soft solids was systematically investigated through theoretical analysis and finite element method (FEM). The validation and application of the Hertzian load-displacement relation for indentation of hyperelastic soft materials were clarified, the effects of large deformation and material nonlinearity on spherical indentation of hyperelastic soft materials were analyzed and discussed. It was found that the complicated indentation behaviors of hyperelastic soft solids mainly depended on the coupling interactions of large deformation and material nonlinearity. Besides, we proposed two new nonlinear elastic contact models to separate the effects of large deformation and material nonlinearity on spherical indentation responses of hyperelastic soft solids. Our efforts might help to enhance the understanding of hyperelastic indentation problems and provided necessary instructions for the mechanical characterization of hyperelastic soft materials.     

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.     

Axi-symmetric contact on thin compliant coatings

This paper presents a theory for the indentation of a soft thin coating by a rigid body. The coating is assumed to be bonded to a rigid substrate and to behave linearly elastically. A simplifying approximation enables the stresses within the coating, averaged through its thickness, to be determined for particular indenter profiles. The results are shown to be sensitive to the thickness and compressibility of the coating material. Unlike much previous work, the results can be expressed analytically for certain indenter profiles and have been substantiated by experiment. The theory has many useful applications, in particular for situations where the layer acts as a protective coating and for the accurate, in situ and non-destructive measurement of the elastic modulus of the coating material.     

瞬态波在层合板中传播的粘弹比拟法

本文发展了粘弹比拟理论,并将之用于求解半无限空间三层复合材料在垂直层合方向传播的瞬态波问题。对于层合板中应力波的传播问题,寻找到了一等效粘弹体,并用一种较好的Laplace变换的数值反演法求得了等效松弛函数和其它一些必要的辅助函数。用特征线法求得了等效粘弹体的应力和速度,进而得到了三层复合材料中心的应力、速度,进一步就得到了层中任意点的应力和速度。对于一个可由精确理论(射线理论)给出计算结果的问题,将粘弹比拟理论的结果和射线理论的结果进行了比较,结果表明,粘弹比拟理论对三层复合材料的瞬态波传播问题是相当成功的。     

Indentation of pressurized viscoplastic polymer spherical shells

The indentation response of polymer spherical shells is investigated. Finite deformation analyses are carried out with the polymer characterized as a viscoelastic/viscoplastic solid. Both pressurized and unpressurized shells are considered. Attention is restricted to axisymmetric deformations with a conical indenter. The response is analyzed for various values of the shell thickness to radius ratio and various values of the internal pressure. Two sets of material parameters are considered: one set having network stiffening at a moderate strain and the other having no network stiffening until very large strains are attained. The transition from an indentation type mode of deformation to a structural mode of deformation involving bending that occurs as the indentation depth increases is studied. The results show the effects of shell thickness, internal pressure and polymer constitutive characterization on this transition and on the deformation modes in each of these regimes.     

A Discontinuous Elastic Interface Transfer Model of Thin Film Nanoindentation

A new model of thin film indentation that accounted for an apparent discontinuity in elastic strain transfer at the film/substrate interface was developed. Finite element analysis suggested that numerical values of strain were not directly continuous across the interface; the values in the film were higher when a soft film was deposited on a hard substrate. The new model was constructed based on this discontinuity; whereby, separate weighting factors were applied to account for the influence of the substrate in strain developed in the film and vice-versa. By comparing the model to experimental data from thirteen different amorphous thin film materials on a silicon substrate, constants in each weighting factor were found to have physical significance in being numerically similar to the bulk scale Poisson’s ratios of the materials involved. When employing these material properties in the new model it was found to provide an improved match to the experimental data over the existing Doerner and Nix and Gao models. Finally, the model was found to be capable of assessing the Young’s modulus of thin films that do not exhibit a flat region as long as the bulk Poisson’s ratio is known.     

考虑膜厚影响薄膜磁致伸缩系数、泊松比和杨氏模量的同时测量

基于经典层合板理论,建立了一个能同时测量薄膜-基底系统中薄膜的磁致伸缩系数、杨氏模量和泊松比的板模型.以前的研究计算薄膜磁致伸缩系数时,大多假设薄膜的弹性属性与相应的块材一致,由此导致的磁致伸缩系数计算是不准确的.在目前大多数方法中仅仅在使用一个单一的弹性各向同性基底中能够避免这个问题.该文模型在各向异性基底下同样适用,并且不要求薄膜的厚度远远小于基底厚度,因此也能够用来计算磁致伸缩应变和设计微电机械系统和生物微电机械系统.对已有的铁基非晶薄膜的实验数据,在不同磁场强度下,磁致伸缩系数的计算结果与已有模型进行了比较,它们之间的差异得到了解释.同时,还可以得到薄膜的弹性常数.     

Analytical transient phase change heat transfer model of wearable electronics with a thermal protection substrate

As thermal protection substrates for wearable electronics, functional soft composites made of polymer materials embedded with phase change materials and metal layers demonstrate unique capabilities for the thermal protection of human skin. Here,we develop an analytical transient phase change heat transfer model to investigate the thermal performance of a wearable electronic device with a thermal protection substrate.The model is validated by experiments and the finite element analysis(FEA). The ...     

Spontaneous instability of soft thin films on curved substrates due to van der Waals interaction

The linear bifurcation theory is used to investigate the stability of soft thin films bonded to curved substrates. It is found that such a film can spontaneously lose its stability due to van der Waals or electrostatic interaction when its thickness reduces to the order of microns or nanometers. We first present the generic method for analyzing the surface stability of a thin film interacting with the substrate and then discuss several important geometric configurations with either a positive or negative mean curvature. The critical conditions for the onset of spontaneous instability in these representative examples are established analytically. Besides the surface energy and Poisson's ratio of the thin film, the curvature of the substrate is demonstrated to have a significant influence on the wrinkling behavior of the film. The results suggest that one may fabricate nanopatterns or enhance the surface stability of soft thin films on curved solid surfaces by modulating the mechanical properties of the films and/or such geometrical properties as film thickness and substrate curvature. This study can also help to understand various phenomena associated with surface instability.     

Dynamic rheological properties of a fumed silica grease

The thermo-rheological characteristics of a fumed silica lubricating grease in linear and nonlinear oscillatory experiments have been investigated. The material rheological behavior represents a soft solid being thermo-rheologically complex. There is an abnormal temperature dependency in the range of ??10 to 10 °C which is related to the phase transition of the base oil. The dynamic moduli data in linear viscoelastic envelop (LVE) have been modeled using mode-coupling theory (MCT) in the whole temperature range. Two main relaxation mechanisms can be identified through linear and nonlinear viscoelastic properties related to interaction of the primary particle and its neighbor particles as well as a slow relaxation process which represents the escape of this particle from its “cage”. Finally, it is demonstrated that the dominant yielding process in large amplitude oscillatory experiments can be explained based on either particle cage rupture (consistent with MCT framework) or particle “hopping” out of its cage proposed in soft glassy rheology (SGR) model. It will be discussed that the governing mechanism depends on the applied frequency.     

Measurements of Two Independent Viscoelastic Functions by Nanoindentation

Current nanoindentation measurement techniques normally assume that one material function (such as the Poisson's function) is a constant, and measures just one material function, such as the creep compliance in shear. For materials with significant viscoelastic effects and unknown viscoelastic functions, assuming a constant for one material function is not satisfactory. Accurate measurements require simultaneously determining two independent material functions. This paper provides a method to use nanoindentation to measure both bulk and shear relaxation functions. Two different nanoindenter tips, namely Berkovich and spherical indenters, are used for nanoindentation on polymers. Any two independent viscoelastic functions, such as bulk relaxation modulus and shear relaxation modulus, have different representations in the load–displacement curves obtained with these two indenters so that the two independent viscoelastic functions can be separated and determined. Two polymers, poly(vinyl acetate) (PVAc) and poly(methyl methacrylate) (PMMA) were used in nanoindentation. Nanoindentation measurements were conducted on PVAc above glass transition temperature (Tg) and on PMMA below Tg. Both shear and bulk relaxation functions determined from nanoindentation were found in a reasonably good agreement with data obtained from conventional tests, providing validation of the method presented. The new method can be applied in measurements of two independent viscoelastic functions at sub-micron scale of very small amounts of materials such as polymeric films on a substrate, heterogeneous materials such as bones, tissues, and nanocomposites.     

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.     

SIZE EFFECT AND GEOMETRICAL EFFECT OF SOLIDS IN MICRO—INDENTATION TEST

Micro-indentation tests at scales of the order of sub-micron show that the measured hardness increases strongly with decreasing indent depth or indent size,which is frequently referred to as the size effect.At the same time,at micron or sub-micron scale,another effect,which is referred to as the geometrical size effects such as crystal grain size effect,thin flim thickness effect,etc.,also influences the measured material hardness.However,the trends are at odds with the size-independence implied by the conventional elastic-plastic theory.In the present research,the strain gradient plasticity theory(Fleck and Hutchinson)is used to model the composition effects(size effect and geometrical effect) for polycrystal material and metal thin film/ceramic substrate systems when materials undergo micro-indenting.The phenomena of the “pile-up“ and “sink-in“ apeared in the indentation test for the polycrystal materials are also discussed.Meanwhile,the micro-indentation experiments for the polycrystal Al and for the Ti/Si3N4 thin film/substrate system are carried out.By comparing the theoretical predictions with experimental measuremtns.the values and the variation trends of the micro-scale parameter included in the strain gradient plasticity theory are predicted.     

On modeling the micro-indentation response of an amorphous polymer

Interest in instrumented indentation experiments as a means to estimate mechanical properties has grown rapidly in recent years. Although numerous nano/micro-indentation experimental studies on polymeric materials have been reported in the literature, a corresponding methodology for extracting material property information from the experimental data does not exist. This situation for polymeric materials exists primarily because baseline numerical analyses of sharp indentation using appropriate large deformation constitutive models for the nonlinear viscoelastic–plastic response of these materials appear not to have been previously reported in the literature. An existing, widely used theory for amorphous polymers (e.g. [Boyce, M., Parks, D., Argon, A.S., 1988. Large inelastic deformation of glassy polymers. Part 1: Rate dependent constitutive model. Mechanics of Materials 7, 15–33; Arruda, E.M., Boyce, M.C., 1993. Evolution of plastic anisotropy in amorphous polymers during finite straining. International Journal of Plasticity 9, 697–720]) has been recently found to lack sufficient richness to enable one to quantitatively reproduce the major features of the indentation load-versus-depth curves for some common amorphous polymers [Gearing, B.P., 2002. Constitutive equations and failure criteria for amorphous polymeric solids. Ph.D. thesis, Massachusetts Institute of Technology].This study develops a new continuum model for the viscoelastic–plastic deformation of amorphous polymeric solids. We have applied the constitutive model to capture salient features of the mechanical response of the amorphous polymeric solid poly(methyl methacrylate) (PMMA) at ambient temperature and stress states under which this material does not exhibit crazing. We have conducted compression-tension strain-controlled experiments, as well as stress-controlled compression-creep experiments, and these experiments are used to calibrate the material parameters in the constitutive model for PMMA.We have implemented our constitutive model in a finite-element computer program, and using this finite-element program we have simulated micro-indentation experiments on PMMA. We show that our constitutive model and finite element simulations reproduce the experimentally-measured indentation load-versus-depth response with reasonable accuracy.