Indentation of elastic solids with rigid cones

Experimental investigations of the response of several elastic solids, carried out using an instrumented indentation machine, are described. These solids include a natural rubber compound, neoprene, and three different compositions of poly(dimethylsiloxane) (PDMS) when they were indented with tungsten carbide cones of included angles of 60°, 90°, 120° and 136°. It is shown that, except for the case of the 60° cone loading on to the blocks of PDMS (1?:?10) and PDMS (1?:?20), the indentation load versus indenter displacement behaviour of all the elastic solids corresponding to all the conical indenters is well fitted by the 1965 Sneddon theory for frictionless indentations. Video camera sequences of the process of indentation by cones of different included angles into the blocks of PDMS (1?:?10) and PDMS (1?:?20) have also been taken from which measurements of the elastic deformation of the indented surface have been made. It is shown that the magnitude of the elastic deformation of the indented surface closely agrees with the theory when the blocks are indented with cones of included angles of 90°, 120° and 136°. However, for the case of the 60° cone, there is again a discrepancy between experiment and theory; the magnitude of the deformation is significantly smaller than that predicted theoretically. It is suggested that the discrepancy between theory and experimental observations may be due to a non-zero coefficient of friction between the 60° conical indenter and the blocks of PDMS (1?:?10) and PDMS (1?:?20). It is also shown experimentally that, for indenter loading speeds in the range 0.2–200?mm?min^?1, no time-dependent effects were observed for the loading curves, thus justifying the use of the theory given by Love in and of the Sneddon theory of conical indentations of elastic solids. Finally, using the findings described in this work, a brief discussion is included to explain the discrepancy between the results obtained by Sabey in and those obtained by Greenwood and Tabor in the same year when tyre-tread rubber specimens obtained from a single source were indented with steel cones of different included angles.     

Nanoelectromechanics of piezoelectric indentation and applications to scanning probe microscopies of ferroelectric materials

Nanoelectromechanics of piezoelectric indentation, including the structure of coupled electroelastic fields and stiffness relations, is analysed for flat, spherical, and conical indenter geometries. Exact solutions in elementary functions for electroelastic fields inside the material are obtained using the recently established correspondence principle between the elastic and the piezoelectric problems. The stiffness relations fully describe the indentation process and relate indentation depth, indentation force and bias to the relevant material properties and indenter parameters. This extends the results of Hertzian mechanics to piezoelectric materials. The stiffness relations are utilized for quantitative understanding of the electromechanical scanning probe microscopies (SPM) of ferroelectric and piezoelectric materials, including piezoresponse force microscopy, atomic force acoustic microscopy, scanning near-field acoustic microscopy, and heterodyne ultrasonic-electrostatic force microscopy. The structure of the electroelastic field yields a quantitative measure of signal generation volume in electromechanical SPMs and also provides a quantitative basis for the analysis of tip-induced polarisation switching and local hysteresis loop measurements.     

Identification of elastic-plastic and phase transition characteristics for relaxor ferroelectric PMN-PT anisotropic single crystals using nanoindentation technique

As one kind of important ferroelectric ceramics, relaxor ferroelectric PMN-PT single crystals have triggered a revolution in electromechanical devices owing to their giant piezoelectric properties and ultra-high electromechanical coupling factors. The present study focused on the mechanical responses of [100]- and [110]-oriented poled PMN-PT ferroelectric single crystals under an indenter loading. The hardness and Young’s modulus with different crystallographic orientations of the crystals were measured by using the continuous stiffness measurement (CSM) with nanoindentation technique. Using a spherical indenter pressured at different indentation depths, the typical quasi-static nanoindentation tests with displacement-controlled mode were performed on the PMN-PT single crystal samples. Load–displacement curves of indentations were recorded to reveal the yielding or inelasticity behaviour in [100]- and [110]-oriented PMN-PT through a pop-in event. It was further verified by the stress–strain curves evaluated from the corresponding load–displacement curves, to show the similar characteristic on the elastic–inelastic transition. When a Berkovich indenter was employed for mechanical response testing, another pop-in event was observed at a smaller indentation depth compared to the one for elastic–inelastic transition, which may indicate a pressure-induced phase transition from rhombohedral (R) to tetragonal (T) of the PMN-PT single crystals.     

Effects of membrane pre-stress and intrinsic viscoelasticity on nanoindentation of cells using AFM

Nanoindentation using atomic force microscopy (AFM) has found a wide range of applications in characterizing the mechanical properties of cells. However, both conventional Hertz theory and Sneddon's solution face difficulties in interpreting cell indentation data due to lack of considerations of the bilayered structure of cells, the pre-stress of cell membranes and the intrinsic viscoelasticity of cell interior phases. In the present study, the indentation of a cell using AFM is modelled as that of a pre-tensed elastic shell supported by a viscoelastic half-space. Analytical solutions are derived for the shallow indentation of the elastic counterpart of the bilayered structure and then extended to moderate-depth indentation. The cell membrane and its pre-tension are important in interpreting the indentation data if a small indenter is used. Based on the elastic solutions, viscoelastic solutions are derived for creep tests, relaxation tests and linear loading tests, and verified by finite element analysis. Parametric studies were performed to investigate the effects of the membrane pre-stress and the intrinsic viscoelasticity of the cell on the relation between the indentation load vs. indentation depth. In addition, an inverse analysis was performed to extract the viscoelastic parameters of the cell interior phase and the uniqueness of the extraction was assessed.     

Indentation of transversely isotropic power-law hardening materials: computational modelling of the forward and reverse problems

Using a combination of dimensional analysis and large deformation finite element simulations of triple indentations of 120 materials, a framework for capturing the indentation response of transversely isotropic materials is developed. By considering 4800 combinations of material properties within the bounds of the original set of 120 materials, forward algorithms that predict the indentation response of materials and reverse algorithms that predict the materials’ elastic and plastic properties from experimentally measured indentation responses are formulated for both longitudinal and transverse indentations. Issues of accuracy, reversibility, uniqueness and sensitivity within the context of the indentation of transversely isotropic materials are addressed carefully. Using 1400 combinations of material properties, it is demonstrated that there is perfect reversibility between the material properties and their indentation responses as predicted by the forward and reverse algorithms. On average, the differences between the results of the finite element analysis and those predicted by the forward algorithms for longitudinal or transverse indentations are less than 1%, thus demonstrating the high accuracy and uniqueness of the forward analysis. For longitudinal and transverse indentations, the reverse algorithms provide accurate results in most cases with an average error of 3 and 6%, respectively. A sensitivity analysis with a ±2% variation in the material properties in the forward algorithm and ±2% variation in the indentation responses in the reverse algorithms demonstrated the robustness of the algorithms developed in the present study, with the longitudinal indentations providing relatively less sensitivity to variability in indentation responses as compared to the transverse indentations.     

Indentation on one-dimensional hexagonal quasicrystals: general theory and complete exact solutions

This paper presents a general account of the indentation responses of a one-dimensional hexagonal quasicrystal half-space pressed by an axisymmetric rigid punch. Based on Green's functions of the half-space subjected to point sources on the surface, the mixed boundary value problem is transformed to integral equations and solved exactly using the results of the potential theory method. Explicit expressions for the generalised pressures and indentation forces are derived for three common indenters (cylinder, cone and approximate sphere) in a systematic manner. For conical and spherical indenters, relations between the contact radius and indentation loads are determined. The coupling phonon–phason fields in the half-space under indentation are accurately expressed in terms of elementary functions. Numerical calculations are performed and discussions on related physical phenomena are given. The present exact solutions can serve as benchmarks for approximate or numerical analyses and can guide the experimental characterisation of material properties of quasicrystals.     

Deformation field induced by spherical indentation: numerical analysis and experimental measurement by speckle interferometry

The present paper describes an experimental procedure for the measurement of the displacement field around a spherical indentation. The measurements were performed by a speckle interferometer designed for the detection on small areas of one or more oblique components of displacement; the indentations were made by a standard durometer used for metals.The experiments were carried out on a hardened and tempered steel, previously characterized by a standard tensile test; the results are in accordance with the numerical results obtained by an elasto-plastic FEM analysis.     

Analysis of two colliding fractionally damped spherical shells in modelling blunt human head impacts

The collision of two elastic or viscoelastic spherical shells is investigated as a model for the dynamic response of a human head impacted by another head or by some spherical object. Determination of the impact force that is actually being transmitted to bone will require the model for the shock interaction of the impactor and human head. This model is indended to be used in simulating crash scenarios in frontal impacts, and provide an effective tool to estimate the severity of effect on the human head and to estimate brain injury risks. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. It is assumed that the viscoelastic features of the shells are exhibited only in the contact domain, while the remaining parts retain their elastic properties. In this case, the contact spot is assumed to be a plane disk with constant radius, and the viscoelastic features of the shells are described by the fractional derivative standard linear solid model. In the case under consideration, the governing differential equations are solved analytically by the Laplace transform technique. It is shown that the fractional parameter of the fractional derivative model plays very important role, since its variation allows one to take into account the age-related changes in the mechanical properties of bone.     

Geometry independence of the normalized relaxation functions of viscoelastic materials in indentation

The normalized relaxation modulus represents a salient feature of viscoelastic materials and its determination is of great significance for various applications. From the normalized relaxation modulus, for instance, one can derive the loss factor of a viscoelastic polymer and judge whether a material is suitable for damping applications or not. By using dimensional analysis and the elastic–viscoelastic correspondence principle, the normalized relaxation function of a linear viscoelastic material obtained from indentation relaxation tests is shown to depend only on the indentation load but not on the indenter geometry and the shape of the indented solid. The result could enable circumvention of the difficulties encountered in the calibration of the indenter geometry and the preparation of indented samples. Numerical simulations are performed on a number of cases of practical interest, including the spherical indentation test of a soft layer lying on a rigid substrate, a flat punch indenter indenting into a soft layer with a rough surface bonded to a rigid substrate, a rigid indenter with irregular shape indenting into a particle, inclined contact of a cylindrical indenter with a cylinder, and indentation of porous substrates. The numerical examples demonstrate that the conclusion from the theoretical analysis is valid for all these situations.     

Finite element simulation and experimental determination of interfacial adhesion properties by wedge indentation

This paper presents our recent study on determination of interfacial adhesion properties of soft-film-on-hard-substrate (SFHS) systems using finite element simulation (FEM) and wedge indentation experiments. The objectives of this study are: (i) to simulate the interfacial delamination processes during wedge indentation experiments; (ii) to study the effects of interfacial delamination on the characteristics of the indentation load–displacement (P–h) curves, (iii) to determine the interfacial adhesion properties; and (iv) to compare the simulation and experimental results. During the FEM simulation, a traction-separation law is used to describe the interfacial adhesion properties due to the large-scale yielding during indentations. The effects of main parameters in the traction-separation law, i.e. interfacial strength and interfacial energy, to the initiation of interfacial delamination are studied by parametric studies. An interface energy-strength contour, which can be used to determine the interfacial adhesion properties of the thin-film/substrate systems based on a wedge indentation experiment, is developed from the outcomes of the FEM simulation of the indentations using wedge tips with the inclusion angles of 90° and 120°. Using the respective interface energy-strength contours, the interfacial energy and strength of a BlackDiamond® (BD)/Si system and a methylsilsesquioxane (MSQ)/Si system are determined. The simulated results are then compared with the previous experimentally derived interfacial fracture toughness values and some further discussions are given.     

Estimating localized oscillatory tissue motion for assessment of the underlying mechanical modulus

The technique of harmonic motion imaging (HMI) uses the localized stimulus of the oscillatory ultrasonic radiation force as produced by two overlapping beams of distinct frequencies, and estimates the resulting harmonic displacement in the tissue in order to assess its underlying mechanical properties. In this paper, we studied the relationship between measured displacement and stiffness in gels and tissues in vitro. Two focused ultrasound transducers with a 100 mm focal length were used at frequencies of 3.7500 MHz and either 3.7502 (or 3.7508 MHz), respectively, in order to produce an oscillatory motion at 200 Hz in the gel or tissue. A 1.1 MHz diagnostic transducer (Imasonics, Inc.) was also focused at 100 mm and acquired 5 ms RF signals (pulse repetition frequency (PRF)=3.5 kHz) at 100 MHz sampling frequency during radiation force application. First, three 50x50 mm(2) acrylamide gels were prepared at concentrations of 4%, 8% and 16%. The resulting displacement was estimated using crosscorrelation techniques between successively acquired RF signals with a 2 mm window and 80% window overlap at 1260 W/cm(2). A normal 1-D indentation instrument (TeMPeST) applied oscillatory loads at 0.1-200 Hz with a 5 mm-diameter flat indenter. Then, 12 displacement measurements in 6 porcine muscle specimens (two measurements/case, as above) were made in vitro, before and after ablation which was performed for 10 s at 1260 W/cm(2). In all gel cases, the harmonic displacement was found to linearly increase with intensity and exponentially decrease with gel concentration. The TeMPeST measurements showed that the elastic moduli for the 4%, 8% and 16% gels equaled 3.93+/-0.06, 17.1+/-0.2 and 75+/-2 kPa, respectively, demonstrating that the HMI displacement estimate depends directly on the gel stiffness. Finally, in the tissues samples, the mean displacement amplitude showed a twofold decrease between non-ablated and ablated tissue, demonstrating a correspondence between the HMI response and an increase in stiffness measured with the TeMPeST instrument.     

On the uniqueness and sensitivity issues in determining the elastic and plastic properties of power-law hardening materials through sharp and spherical indentation

A systematic study of the uniqueness, reversibility and sensitivity issues associated with seven indentation-based methods of property extraction demonstrates that: (i) The indentation algorithms generally identify the elastic and plastic properties of materials uniquely for most materials. (ii) The indentation forward algorithms (wherein the indention responses are determined from the elastic and plastic properties of the indented materials) and the reverse algorithms (wherein the elastic and the plastic properties of materials are extracted from the indentation responses) are distinct for each indentation method and are internally consistent in that the differences in the elastic and plastic properties determined through the reverse analysis and the ‘true’ material properties are generally small for a large number of materials, for each of the seven methods. (iii) While the differences in the indentation response parameters predicted by each of the seven indentation methods (for a particular material) could be small, there could be considerable dispersion in the elastic and plastic properties predicted by the reverse algorithms of the seven methods (for a particular set of indentation response parameters). (iv) In the forward analysis, small uncertainties in the elasto-plastic properties lead to small uncertainties in the predictions of the indentation response of materials. The sensitivity distribution is generally heterogeneous and symmetric across positive and negative variations in the material elasto-plastic properties. (v) In the reverse analysis, the elastic modulus exhibits low sensitivity, while the yield strength and the strain-hardening exponent generally exhibit high sensitivity to uncertainties in the indentation response parameters. The sensitivity distribution is heterogeneous and asymmetric across positive and negative variations in the indentation response parameters. (vi) The representative stresses are fairly robust to uncertainties in the indentation response parameters. Consequently, dual sharp and spherical indentation methods, which identify multiple representative stresses, exhibit reduced sensitivity in the determination of the plastic properties.     

Probe Indentation: A Mesoscale Approach to Characterise Powder Systems: Experimental Investigation of Monomodel and Bimodal Diameter Distributions of Glass Spheres

An experimental study investigating a novel power characterisation method is investigated. The scope of this experimental work is to assess the feasibility, suitability and sensitivity of small scale probe indentation as a mechanism by which Discrete element method (DEM) may be calibrated. This meso–scale approach is chosen for investigation since it bridges the gap between the single‐particle methodologies of atomic force microscopy (AFM) and bulk measurements such as shear cell testing. Five different mono‐modal populations of glass bead and bi‐modal blends of these populations have been examined. The force‐displacement profile from a spherical probe was measured during indentation into a powder bed. Discernable differences in the resultant force‐profile are quantifiable and a mechanism has been proposed for the physical basis of these different characteristics.     

Wave equations in linear viscoelastic materials

Conditions for writing wave equations in linear viscoelastic materials are investigated. The study is restricted to the infinitesimal theory and an application is suggested in modeling ultrasound propagation in soft biological tissues. First, a general wave equation is obtained for the displacement field in an inhomogeneous medium. Second, the propagation of "the mean principal stress" (i.e., minus the arithmetical mean of the principal stresses) is examined. That quantity is particularly relevant when the force per unit area is detected at the surface of a nondissipative coupling medium. If the material is homogeneous, a wave equation is always obtained for the mean principal stress. Otherwise, supplementary conditions have to be assumed on the material and possibly on the motion. Results are illustrated by examples which present linearly elastic perfect fluids and linearly elastic Newtonian viscous fluids as particular viscoelastic materials.     

Comparison of the surface wave method and the indentation method for measuring the elasticity of gelatin phantoms of different concentrations

The speed of the surface Rayleigh wave, which is related to the viscoelastic properties of the medium, can be measured by noninvasive and noncontact methods. This technique has been applied in biomedical applications such as detecting skin diseases. Static spherical indentation, which quantifies material elasticity through the relationship between loading force and displacement, has been applied in various areas including a number of biomedical applications. This paper compares the results obtained from these two methods on five gelatin phantoms of different concentrations (5%, 7.5%, 10%, 12.5% and 15%). The concentrations are chosen because the elasticity of such gelatin phantoms is close to that of tissue types such as skin. The results show that both the surface wave method and the static spherical indentation method produce the same values for shear elasticity. For example, the shear elasticities measured by the surface wave method are 1.51, 2.75, 5.34, 6.90 and 8.40 kPa on the five phantoms, respectively. In addition, by studying the dispersion curve of the surface wave speed, shear viscosity can be extracted. The measured shear viscosities are 0.00, 0.00, 0.13, 0.39 and 1.22 Pa.s on the five phantoms, respectively. The results also show that the shear elasticity of the gelatin phantoms increases linearly with their prepared concentrations. The linear regressions between concentration and shear elasticity have R² values larger than 0.98 for both methods.     

A simple-shear rheometer for linear viscoelastic characterization of vocal fold tissues at phonatory frequencies

Previous studies reporting the linear viscoelastic shear properties of the human vocal fold cover or mucosa have been based on torsional rheometry, with measurements limited to low audio frequencies, up to around 80 Hz. This paper describes the design and validation of a custom-built, controlled-strain, linear, simple-shear rheometer system capable of direct empirical measurements of viscoelastic shear properties at phonatory frequencies. A tissue specimen was subjected to simple shear between two parallel, rigid acrylic plates, with a linear motor creating a translational sinusoidal displacement of the specimen via the upper plate, and the lower plate transmitting the harmonic shear force resulting from the viscoelastic response of the specimen. The displacement of the specimen was measured by a linear variable differential transformer whereas the shear force was detected by a piezoelectric transducer. The frequency response characteristics of these system components were assessed by vibration experiments with accelerometers. Measurements of the viscoelastic shear moduli (G' and G") of a standard ANSI S2.21 polyurethane material and those of human vocal fold cover specimens were made, along with estimation of the system signal and noise levels. Preliminary results showed that the rheometer can provide valid and reliable rheometric data of vocal fold lamina propria specimens at frequencies of up to around 250 Hz, well into the phonatory range.     

TiN薄膜在纳米压痕和纳米划痕下的断裂行为

利用磁控溅射方法在Si(111)衬底上制备了具有(111)和(222)择优取向的TiN薄膜. 用纳米压痕和纳米划痕方法研究了该薄膜的变形和断裂行为. 用扫描电子显微镜、纳米压痕原位原子力显微镜及原位光学显微镜并结合加-卸载 曲线及划痕曲线获得了薄膜发生变形和断裂的微观信息. 在压痕试验中, TiN薄膜在压入深度为200 nm时表现为塑性变形及压痕周围的局部断裂, 随着压入深度的增大, 塑性变形和局部断裂变得越显著, 当最大压入深度达到临界值1000 nm时, 薄膜和衬底间发生了界面断裂. 在划痕实验中, 100 mN及200 mN的最大载荷均可以引起界面断裂. 最大为200 mN的载荷使得薄膜发生界面断裂的位置比用100 mN载荷时的位置提前, 但其临界断裂载荷和100 mN时及压痕实验时的临界界面断裂载荷基本相同. 关键词： TiN薄膜 纳米压痕 纳米划痕 界面断裂     

Evaluation of adhesive and elastic properties of materials by depth-sensing indentation of spheres

Work of adhesion is the crucial material parameter for application of theories of adhesive contact. It is usually determined by experimental techniques based on the direct measurements of pull-off force of a sphere. These measurements are unstable due to instability of the load-displacement diagrams at tension, and they can be greatly affected by roughness of contacting solids. We show how the values of work of adhesion and elastic contact modulus of materials may be quantified using a new indirect approach (the Borodich?CGalanov (BG) method) based on an inverse analysis of a stable region of the force-displacements curve obtained from the depth-sensing indentation of a sphere into an elastic sample. Using numerical simulations it is shown that the BG method is simple and robust. The crucial difference between the proposed method and the standard direct experimental techniques is that the BG method may be applied only to compressive parts of the force-displacements curves. Finally, the work of adhesion and the elastic modulus of soft polymer (polyvinylsiloxane) samples are extracted from experimental load-displacement diagrams.     

Direct numerical simulation of elastic turbulence and its mixing-enhancement effect in a straight channel flow

In this paper,we present a direct numerical simulation(DNS) of elastic turbulence of viscoelastic fluid at vanishingly low Reynolds number(Re = 1) in a three-dimensional straight channel flow for the first time,using the Giesekus constitutive model for the fluid.In order to generate and maintain the turbulent fluid motion in the straight channel,a sinusoidal force term is added to the momentum equation,and then the elastic turbulence is numerically realized with an initialized chaotic velocity field and a stretched conformation field.Statistical and structural characteristics of the elastic turbulence therein are analyzed based on the detailed information obtained from the DNS.The fluid mixing enhancement effect of elastic turbulence is also demonstrated for the potential applications of this phenomenon.     

Calculation of the acoustic radiation force on coated spherical shells in progressive and standing plane waves

In this paper, analytical equations are derived for the time-averaged radiation force induced by progressive and standing acoustic waves incident on elastic spherical shells covered with a layer of viscoelastic and sound-absorbing material. The fluid surrounding the shells is considered compressible and nonviscous. The incident field is assumed to be moderate so that the scattered field from the shells is taken to linear approximation. The analytical results are illustrated by means of a numerical example in which the radiation force function curves are displayed, with particular emphasis on the coating thickness and the content of the hollow region of the shells. The fluid-loading on the radiation force function curves is analysed as well. This study attempts to generalize the various treatments of radiation force due to both progressive and standing waves on spherically-shaped structures immersed in ideal fluids. The results show that various ways can be effectively used for damping resonance peaks, such as by changing the fluid in the interior hollow region of the shells or by changing the coating thickness.