Refined model of elastic nanoindentation of a half-space by the blunted Berkovich indenter accounting for tangential displacements on the contact surface

Elastic contact between a non-ideal Berkovich indenter and a half-space is investigated. The derived mathematical model of the contact allows for tangential displacements of the boundary points of the half-space. The tip of the blunted indenter is simulated as a smooth surface. The boundary element method is implemented in the model for numerical simulation of nanoindentation. The relative deviation function is introduced and calculated to quantify the influence of the tangential displacements on the load–displacement curves. A simple expression is derived for the impact of the tangential displacements on the values of the reduced Young’s modulus determined due to nanoindentation studies. The refined model was successfully applied to simulate the experimental load–displacement curves gained by elastic nanoindentations of flat LiF and KCl samples. Such values of the indenter bluntness (the varying parameter) were found that the simulated load–displacement curves coincided with those of the experimental data at displacements higher than 7.5 nm. The model neglecting tangential displacements gives slightly differing values for the parameter of the indenter bluntness.

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Determination of the hardness and elastic modulus from continuous vickers indentation testing

Continuous Vickers (H _v) indentation tests were performed on different materials (ion crystals, metals, ceramics, silica glass and plastic). Load-indentation depth curves were taken during the loading as well as during the unloading period by a computer controlled hydraulic mechanical testing machine (MTS 810). The indentation work measured both the loading and the unloading periods, and these were used for the evaluation of parameters characterizing the materials. It was found empirically that there were linear connections between the maximum load to the power 3/2 and the indentation work. These connections were used to relate the conventional hardness number, H _v, and Young's modulus, E, with the work performed during loading and unloading. This work can be determined with great accuracy from the measurements. The values of the Young's modulus and the Vickers hardness determined this way agree well with those obtained by conventional methods. On the basis of continuous indentation tests, materials can be easily classified into the isomechanical groups introduced by Ashby. For this classification the H _v/E ratio is generally used. As a substitute for H _v/E another parameter is recommended which can be determined easily from a single measurement.     

Defect of the elastic modulus of composition materials under ultrasound loading

The effect of the ultrasound loading rate and the amplitude-cyclic and thermal action conditions on the behavior of the Young modulus defect for a Fe – 30% Cu iron-copper pseudo alloy and its components is determined. It is demonstrated that the basic mechanisms of internal energy dissipation in composition materials manifest thermsleves in stages; the characteristics of these mechanisms are analyzed.Translated from Problemy Prochnosti, No. 10, pp. 32–36, October, 1992.     

Representative indentation elastic modulus evaluated by unloading of nanoindentation made with a point sharp indenter

The conventional method to extract elastic modulus from the nanoindentation on isotropic linearly elastic solids is based on Sneddon’s solution (1965). However, it is known that the solution is valid only for incompressive elastic solids with the Poisson’s ratio ν of 0.5. This paper first proposes the modification of the solution in a wide range of ν from 0 to 0.5 through the numerical analysis on the unloading behavior of a simulated conical nanoindentation with a finite element method. As a result of the modification, the coefficient of linearity between the indentation elastic parameter k_e and Young’s modulus E is empirically given as a function of ν and the inclined face angle of the indenter, β, where k_e is defined as k_e ≡ P/h² with the indentation load P and penetration depth of the indenter h. According to the linear relationship between k_e and E, it is found that elastic rebound during unloading of a nanoindentation is uniquely characterized by a representative indentation elastic modulus E¹ defined in terms of E, ν and β, and that the value of E¹ can be evaluated from the P–h relationship with k_e and β. For an isotropic elastoplastic solid, the indentation unloading parameter k₂ defined as k₂ ≡ P/(h–h_r)² for a residual depth h_r is different from k_e even though a linearly elastic solid with k_e and elastoplastic solid with k₂ have a common E¹. In order to evaluate E¹ of an elastoplastic solid, the corresponding k_e is estimated from k₂ with an empirical equation as a function of the relative residual depth ξ defined as ξ ≡ h_r/h_max for the maximum penetration depth h_max. A nanoindentation experiment confirmed the validity of the numerical analysis for evaluating the elastic modulus.     

Determination of elastic modulus in a nickel alloy from ultrasonic measurements

Elastic constants relate technological, structural and safety aspects to various materials phenomena and to their fundamental interatomic forces. Hence, they are of fundamental importance in almost all engineering applications. Thus its determination is of utmost importance. The aim of the present investigation is to study the behaviour of elastic constants and the variation on heat treatment in a nickel base super alloy Nimonic 263 by ultrasonic velocity measurements. From the present study it is evident that the elastic moduli of the material are very sensitive to any minor compositional changes, resulting due to the formation of intermetallic phases on heat treatment and can be effectively monitored by ultrasonic.     

Evaluation of the elastic modulus of thin film considering the substrate effect and geometry effect of indenter tip

A method using finite element method (FEM) is proposed to evaluate the geometry effect of indenter tip on indentation behavior of film/substrate system. For the nanoindentation of film/substrate system, the power function relationship is proposed to describe the loading curve of the thin film indentation process due to substrate effect. The exponent of the power function and the maximum indentation load can reflect the geometry effect of indenter and substrate effect. In the forward analysis, FEM is used to simulate the indentation behavior of thin film with different apex angles of numerical conical indenter tip, and maximum indentation load and loading curve exponent are obtained from the numerical loading curves. Meanwhile, the dimensionless equations between the loading curve exponent, the maximum load, elastic properties of film/substrate system and apex angle of indenter are established considering substrate effect. In the reverse analysis, a nanoindentation test was performed on thin film to obtain the maximum indentation load and the loading curve exponent, and then the experimental data is substituted into the dimensionless equations. The elastic modulus of thin film and the real apex angle of indenter can be obtained by solving the dimensionless equations. The results can be helpful to the measurement of the mechanical properties of thin films by means of nanoindentation.     

Determination of the flexural strength and elastic modulus of ice from in situ cantilever-beam tests

From the theory of cantilever beams on an elastic foundation, it is shown that the strength index and modulus index of ice can be determined from measurements of either the failure load or the tip deflection, or both, of in situ cantilever beams tested over a wide enough range of ratio of beam length to beam thickness. Four methods are proposed, two of which do not require the measurement of beam deflection during beam loading, an often difficult task to perform with sufficient reliability, especially in the field. The methods have been applied to available field and laboratory data. The initial results show reasonably consistent estimates of the strength index but a large variation in the predicted values of the modulus index. One preferred method is suggested but its validity and reliability need to be further evaluated by analyzing a sufficient number of field and laboratory tests.     

Determination of hardness to elastic modulus ratios using Knoop indentation measurements and a model based on loading and reloading half-cycles

A model is developed which results in an expression relating the residual in-surface dimension of the minor diagonal of a Knoop indenter through a residual-width parameter (b _R/b^*)² to the hardness-to-modulus ratio (H/E) of elastic-plastic materials. The relationship is shown to predict within reasonable accuracy both the intercept and slope of a plot of (b _R/b^*)² against (H/E), using data for a variety of materials. The work supplements concepts presented in two previous investigations.     

Calculating the elastic modulus from nanoindentation and microindentation reload curves

The Oliver–Pharr method was used to calculate the elastic modulus from the reloading curve and was compared to the traditional unloading curve method. Nanoindentation and microindentation testing instruments were used. This method was applied to load–unload–reload–unload, multistep, and cycle indentation testing procedures at various hold times and force rates. On unloading the reverse plasticity added to the elastic recovery which increased the apparent elastic modulus. During reloading there was mainly elastic deformation making it more reliable for the elastic modulus calculation. It was also found that the metals tested started yielding between 70% and 100% of the reload curve. The reload indentation elastic modulus for fused silica and several metals was equivalent to the tensile test elastic modulus from reference literature.     

Determination of the elastic modulus of adherent cells using spherical atomic force microscope probe

When the conventional Hertz formula is used to extract the elastic modulus, E, of cells based on the compression test using atomic force microscope spherical probe, the inconsistency between the actual situation and the assumption of the formula will lead to a large error. Using the ABAQUS for finite element modeling and analysis, here, a modified Hertz formula was developed to reduce the effects of cell radius, cell thickness, probe radius and compression depth on the extracted E of cells. Experimentally, the insensitivity of the extracted E to the compression region of cell and probe radius reflects the validity of the modified formula. Owing to the poor resolution of spherical probes, it's unlikely to know the actual thickness of cell at the measured point, which can lead to a huge error. Based on the modified formula, we further proposed an approach to control the effect of the uncertainty of cell thickness and ensured that a 10% difference in cell thickness does not incur over 10% variation in the obtained elastic modulus.

     

Size effect of indenter on determining modulus of nanowires using nanoindentation technique

This study uses a finite element modeling to simulate the behavior of copper nanowires submitted to nanoindentation and estimate their mechanical properties. The simulation results reveal that using the well-known Oliver-Pharr theory, generally applied for materials with semi-infinite half-space, yields an underestimate elastic modulus of the wire materials. Moreover, the radius of the indenter tip also influences the accuracy of the predicted elastic modulus. Such errors are mainly from the overestimate of the contact area between the indenter and the specimen. They can be corrected from the numerical modeling. The elastic modulus of the wires calculated from the corrected contact area is highly close to the bulk value.     

On the elastic modulus of metallic nanowires

Previous atomistic simulations and experiments have attributed size effects in the elastic modulus of Ag nanowires to surface energy effects inherent to metallic surfaces. However, differences in experimental and computational trends analyzed here imply that other factors are controlling experimentally observed modulus changes. This study utilizes atomistic simulations to determine how strongly nanowire geometry and surface structure influence nanowire elastic modulus. The results demonstrate that although these factors do influence the elastic modulus of Ag nanowires to some extent, they alone are insufficient to explain current experimental trends in nanowire modulus with decreasing dimensional scale. Future work needs to be done to determine whether other factors, such as surface contaminants or oxide layers, contribute to the experimentally observed elastic modulus increase.     

Weibull modulus of nano-hardness and elastic modulus of hydroxyapatite coating

Here we report the microstructural dependence of nano-hardness (H) and elastic modulus (E) of microplasma sprayed (MIPS) 230 μm thick highly porous, heterogeneous hydroxyapatite (HAP) coating on SS316L. The nano-hardness and Young’s modulus data were measured on polished plan section (PS) of the coating by the nanoindentation technique with a Berkovich indenter. The characteristic values of nano-hardness and Young’s modulus were calculated through the application of Weibull statistics. Both nano-hardness and the Young’s modulus data showed an apparent indentation size effect. In addition, there was an increasing trend of Weibull moduli values for both the nano-hardness and the Young’s modulus data of the MIPS-HAP coating as the indentation load was enhanced from 10 to 1,000 mN. An attempt was made in the present work, to provide a qualitative model that can explain such behavior.     

The elastic modulus of aluminium-lithium alloys

Young's modulus measurements have been made on Al-Li alloys containing up to 32 at % lithium, in an attempt to determine the cause of the high modulus that characterizes this potentially important alloy system. In alloys of commercial interest (7–11 at %, 2–3 wt % lithium) the modulus is in the range 79 to 83 GPa, the actual value depending on heat-treatment conditions. The major contribution to this increased modulus arises from lithium in solid solution. The Young's moduli of the Al₃ Li and AlLi intermetallic phases are estimated to be 96 GPa and 105 GPa respectively. Additions of magnesium to the Al-Li system produce a small decrease of the modulus, e.g. 4.5 at % (4 wt %) magnesium reduces the modulus by approximately 2 GPa.     

The influence of elastic mismatch between indenter and substrate on Hertzian fracture

This paper is concerned with indentation testing of brittle materials. It is shown that a mismatch of elastic constants between the indenter and component being tested has a profound influence on the stress state induced. It is shown that the development of surface flaws into cracks is severely impeded if the indenter is more rigid than the substrate and vice versa. A quantitative assessment of this phenomenon, in terms of both the contact stress field generated and the stress intensity factor experienced by defects, is given. This permits a more precise determination of either the fracture toughness or the surface flaw distribution to be made.     

Evolution of elastic modulus in roll forming

Roll forming is a continuous process in which a flat strip is incrementally bent to a desired profile. This process is increasingly used in automotive industry to form High Strength Steel (HSS) and Advanced High Strength Steel (AHSS) for structural components. Because of the large variety of applications of roll forming in the industry, Finite Element Analysis (FEA) is increasingly employed for roll forming process design. Formability and springback are two major concerns in the roll forming AHSS materials. Previous studies have shown that the elastic modulus (Young’s modulus) of AHSS materials can change when the material undergoes plastic deformation and the main goal of this study is to investigate the effect of a change in elastic modulus during forming on springback in roll forming. FEA has been applied for the roll forming simulation of a V-section using material data determined by experimental loading-unloading tests performed on mild, XF400, and DP780 steel. The results show that the reduction of the elastic modulus with pre-strain significantly influences springback in the roll forming of high strength steel while its effect is less when a softer steel is formed.     

Modeling of the elastic modulus evolution in unloading-reloading stages

It was already stated that springback in sheet metal forming strongly depends on the elastic properties. Several experimental investigations have revealed that the elastic modulus decreases as the plastic strain increases. Two approaches have been separately employed to explain this phenomenon: dislocations rearrangements and damage. These approaches are considered in a proposed elastoplastic model coupled with damage based on Lemaitre type isotropic ductile damage law. In addition, a hysteresis aspect, which is experimentally observed during unloading-reloading stages, is also considered. Uniaxial tension tests have been used and the predicted results agree well with published experimental data. The proposed model is intended to be implemented in FEM codes for reliable results in forming processes including springback.     

考虑不均匀界面时混凝土弹性模量预测

提出了考虑不均匀界面时混凝土弹性模量预测的解析法。根据界面层上水泥颗粒的分布特性, 给出了界面层上任一点处的局部水灰比和孔隙率。将不均匀界面层划分成一系列同心球壳单元, 通过反演方法确定了每个球壳单元和水泥石基体的弹性模量。将三相混凝土分解成一系列两相复合子结构, 应用两相复合球模型的正确解导出混凝土弹性模量。通过与文献中的两组实验结果比较验证了本文方法的有效性。数值结果表明, 对于给定的骨料体积分数, 混凝土弹性模量随着最大水泥颗粒直径和水灰比的增大而减小, 但随着最大骨料直径的增大而增大, 骨料级配对混凝土弹性模量也有一定的影响。