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.     

An elasto-plastic contact model applied to nanoindentation

This paper is devoted to the finite element modeling of the nanoindentation problem. The frictional contact between the Berkovitch indenter and the very thin elasto-plastic film is treated by the bi-potential method. The elasto-plastic constitutive equation is integrated by means of the radial return mapping algorithm and the consistent tangent operator is explicitly derived. Numerical results show the validity of the model.     

Ab initio study on geometrical structures of the TTTA molecular crystal

The crystal made of organic radical, 1,3,5-trithia-2,4,6-triazapentalenyl (TTTA), exhibits a large first-order magnetic and structural phase transition between a paramagnetic high-temperature (HT) phase and a diamagnetic low-temperature (LT) phase, with a surprisingly wide thermal hysteresis loop over room temperature. We investigate theoretically the crystal structures for the two phases by means of the ab initio structural optimization method based on the local density approximation. We present necessary and sufficient atomic coordinates in this paper. The resulting structures are in good agreement with the recent X-ray diffraction experiment. In the HT phase, uniform one-dimensional stacking of the radical molecule appears, while, in the LT phase, strong dimerization along the stacking direction appears.     

Ab initio study on geometrical structures of the TTTA molecular crystal

The crystal made of organic radical, 1,3,5-trithia-2,4,6-triazapentalenyl (TTTA), exhibits a large first-order magnetic and structural phase transition between a paramagnetic high-temperature (HT) phase and a diamagnetic low-temperature (LT) phase, with a surprisingly wide thermal hysteresis loop over room temperature. We investigate theoretically the crystal structures for the two phases by means of the ab initio structural optimization method based on the local density approximation. We present necessary and sufficient atomic coordinates in this paper. The resulting structures are in good agreement with the recent X-ray diffraction experiment. In the HT phase, uniform one-dimensional stacking of the radical molecule appears, while, in the LT phase, strong dimerization along the stacking direction appears.     

Identification of material model of TiN using numerical simulation of nanoindentation test

The development of the model of the multistep nanoindentation test with Berkovich indenter, accounting for the residual stress distribution, is one of the aims of the present paper. The specimen is unloaded in the intervals between the deformation steps. Substrate, which is composed of a ferritic steel and biocompatible pulsed laser deposition TiN coating, is considered. The selection of the TiN was inspired by its perspective application as the coating for a constructional element of the heart prosthesis (blood chamber and aortic valves). Sensitivity analysis of the model predictions with respect to its parameters is presented in the present paper. The theory of elastic-plastic deformations is used in the finite element model, which simulates both loading and unloading phases, accounting for the real geometry of the indent. The main goal of the present paper was to inversely analyse the tests for coating/substrate system. Square root error between measured and predicted forces is the objective function in the analysis. Results of the inverse calculations, which are presented in the present paper, may be helpful in simulations of the behaviour of TiN deposited on substrate in various applications as bionanomaterials.     

Cross-sectional nanoindentation for copper adhesion characterization in blanket and patterned interconnect structures: experiments and three-dimensional FEM modeling

Cross-sectional nanoindentation (CSN) is a recent method for adhesion measurement of nanoscale thin films in ultra-large scale integrated circuits. In the case of ductile thin films, plastic deformation during the test and complex geometry of delaminated areas require 3D finite element modeling (FEM) for adhesion energy calculation. In this paper the adhesion of various copper (Cu) films on blanket and patterned structures is studied by CSN test. The experimental procedure and qualitative analysis of the test are presented in detail. Crack propagation is studied on blanket and patterned substrates. The dimensions of delaminated blisters are measured by scanning electron microscope (SEM) for each sample. Results show that a geometrical ratio can be used to give a quick and qualitative measurement of adhesion. A new 3D FEM model is then proposed to assess quantitative analysis of CSN test. The deformation energy of Cu blister is calculated for each sample. The mechanical properties of the Cu films required for numerical calculations are measured by instrumented indentation. The influence on these measurements of the evolution of the Cu deformation with penetration depth is discussed in detail with the aid of 2D numerical simulation. The results of numerical modeling correlate well with qualitative evaluation of adhesion.     

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.     

磷酸二氢钾(KDP)晶体纳米压痕过程的有限元分析

为了求得KDP晶体的应力-应变曲线以及材料的屈服应力,基于圣维南定理和实验得到的材料性能参数建立了KDP晶体的压痕过程仿真模型,利用ABAQUS有限元分析软件对KDP晶体压痕过程进行了有限元仿真,得到了KDP晶体的载荷-位移曲线和加/卸载过程中的等效应力变化规律.仿真结果表明:加载过程中最大应力集中在压头尖角处,卸载后最大应力分布在压头棱边所留下的压痕处,KDP晶体材料的屈服应力为120MPa.     

Development of semi-empirical formulation for extracting materials properties from nanoindentation measurements: Residual stresses, substrate effect, and creep

Nanoindentation is one of the most popular techniques for characterizing the mechanical properties of micro- or nano-structured metals or dielectric thin films. However, the obtained experimental data can only provide the relationship between the applied load and the penetration depth. Mechanics models are therefore required to convert the test data into the corresponding material properties. In this work, the effect of residual stress, the substrate effect, and the creep of materials subjected to the indentation test are discussed in order to establish appropriate conversion formulas or criteria for extracting the interested material properties. Dimensional analyses are firstly performed to find the governing parameters and to obtain scaling relationships for subsequent finite element analysis. With the described procedure, models have been developed to convert nanoindentation test data into the desired material properties. Those models provide useful tools for extracting specific material properties, such as residual stress, creep exponent, and stress relaxation time constant. Specifically, this investigation also shows that for the situation of soft film/hard substrate combination, the indentation behavior is essentially identical if the modulus of the substrate is 10 times higher than that of the corresponding film and the response deviates consistently from that of bulk material with increasing of indentation depth. For penetration depth less than 10% of the film thickness, the deviation could be acceptable. On the other hand, significant deviation is observed for hard film/soft substrate systems. In summary, by integrating the models proposed by this work and data from standard tests, it is possible to obtain the Young's modulus, hardness, and the viscoelastic properties as well as the residual stress for a specific material through indentation characterization.     

Alternative methods to determine the elastoplastic properties of sintered hydroxyapatite from nanoindentation testing

This study introduces alternative methods to determine the elastoplastic properties of bovine-derived Hydroxyapatite (HA) porous bone graft through a set of nanoindentation tests with a Berkovich indenter. Generally, experimental data obtained from nanoindentation tests are force displacement, hardness and elastic modulus. However, to determine plastic properties such as strength coefficient and work hardening exponent of bovine HA, analytical or inverse finite element models are required. In this paper, the effect of sintering temperature on these properties of HA is studied for the range of 1000–1400 °C. The direct and inverse Finite Element (FE) simulation models for nanoindentation tests were written in MSC, MARC^® software. A special algorithm for the inverse technique was developed to infer the most suitable elastoplastic material model for HA. A semi-empirical method was adapted to calculate the elastoplastic material properties of HA. The numerical results of harder hydroxyapatite showed better agreement with the experiments while the work hardening exponent, or n-value, and strength coefficient k of hard HA were found to be 0.23 and 8.05 GPa respectively. A comparison between the experimental and predicted load–displacement curves showed that the proposed inverse technique is effective in predicting the elastoplastic material properties from the nanoindentation test with error below 4% at maximum load.     

Nanoindentation of hard multilayer coatings: Finite element modelling

Stress concentrations undermine the load-bearing ability of superhard TiSiN coatings. Experimental studies have shown that multilayer coatings that contain TiSiN layers alternating with ceramic layers with dissimilar mechanical properties suppress contact damage during nanoindentation. In this work, nanoindentation-induced deformation in TiSiN-based multilayer coatings was simulated by means of finite element modelling (FEM). Stress distributions under moderate indentation loading in the structure were quantified with particular emphasis on the relationship between stress concentrations and crack initiation. The results showed that the structural layering can be used to modify the stress distribution, and lower the overall stress level within the coating. In the case of radial tensile stresses at the coating/substrate interface, a reduction ∼50% has been achieved through layering. The resistance to shear damage can also be improved by optimising the multilayer structure.     

Taguchi-based design of experiments in training POD-RBF surrogate model for inverse material modelling using nanoindentation

This paper investigates the use of a surrogate model based on Proper Orthogonal Decomposition (POD) and Radial Basis Functions (RBF) for calibrating the nanoindentation-based loading of an elastic–plastic material. Using Taguchi design of experiments and Analysis of Variance (ANOVA), the total number of finite element-based training points is reduced for input parameters that exhibited lower significance. It is found that ANOVA-based sensitivity information can be used to reduce the number of training points without significantly affecting model accuracy. It is also observed that RBFs capable of conforming nonlinearly perform better when the spatial distance between training points increases. Furthermore, for some RBFs the performance is further tuned by choice of the shape parameters. Finally, it is demonstrated that the surrogate model’s performance remains stable under the effect of random noise. Thereby, this study provides a general framework for solving a nanoindentation-based material modelling inverse problem using the POD–RBF technique.     

Microstructural evolution of copper clad steel bimetallic wire

We investigated the microstructure of two different bimetallic wires of Copper Clad Low Carbon Steel Wire (LCSW), which had a 1006 steel core, and Copper Clad High Carbon Steel Wire (HCSW), which had a 1055 steel core. The HCSW generally showed higher hardness than LCSW because of the pearlitic grain structure. A low temperature annealing at 720 °C to the drawn HCSW caused a significant reduction of hardness, which was as low as that of an annealed LCSW. In general, both LCSW and HCSW showed strong global textured features after drawing, with the steel having a strong 〈1 1 0〉 fiber texture and the copper having a 〈1 1 1 〉-〈1 1 2〉 deformation direction. At the interface, a grain size discrepancy at the steel-copper interface was observed. Post-drawing, the LCSW copper grains exhibited refined grain sizes near the interface and has been explained in terms of shear strain gradient. The HCSW did not exhibit this copper grain size distribution but did exhibit a coarsening of the steel grains near the interface after a subsequent 720 °C heat treatment. This is attributed to the large localized stress concentration at the perimeter of the steel region during the drawing process. The strain induced regions at the steel-copper interface have been simulated by finite element modeling. These grain size discrepancies caused the smooth variation in nanohardness across the interface.     

A theoretical study on the second-order nonlinear optical susceptibilities of lithium formate monohydrate crystal, HCOOLi·H2O

Ab initio calculations of the first hyperpolarizabilities of (HCOOLi·H₂O)_2n supermolecules, as the building-blocks of lithium formate monohydrate (LFM) crystal with extended system, were performed for the first time. The dependence of the static β_ijk⁰ values on chain length was explored, and the frequency dependence of β_ijk(−2ω;ω,ω) was measured, and the influences of electron correlation and basis set on β_ijk⁰ were evaluated. Finally, we predicted the second-order nonlinear optical coefficients of LFM crystal. The β_ijk⁰ value of (HCOOLi·H₂O)_2n is linearly dependent on the chain length of supermolecule, which is quite unusual for an extended system connected by the O–Li bonds with ionic characters. Although the static component of β_zzz⁰ tensor is the static largest in these three components under study, the absolute value of frequency-dependent β_zyy(−2ω;ω,ω) element, transforming the smallest into the largest, is the most sensitive to frequency. After the fundamental wavelength is smaller than 500 nm, it is found that the β_ijk(−2ω;ω,ω) value is resonantly enhanced to a great extent due to the double frequency lies in the region of resonance. In addition, the β_zxx⁰ value goes from negative to positive with changes of electron correlation and basis set. Obviously, it is very necessary to take into account the effect of electron correlation, if the hyperpolarizability tensor components must be accurately calculated. Moreover, it is also very important whether it is adopted a complete basis set with diffuse and polarization functions. The calculated nonlinear coefficients at high level suggest that the scaled set reported by Robert seem more reasonable.     

Effect of superimposed ultrasound on mechanical properties of copper

Abstract

Cu wire is increasingly used in microelectronics packaging manufacturing. In the present paper, the deforming procedure of 100 μm diameter Cu free air balls (FABs) is studied using experimental and finite element (FE) methods. Experimental results show that the superimposed ultrasound can make the Cu FAB softer. Then a non-linear FE model is built to study the deforming process of the Cu FAB. Numerical experiments have been carried out to quantify the effects of superimposed ultrasound on the constitutional model of Cu FABs. An improved power law material model is then established to take into the dynamic acoustic softening effects. A good consistency is achieved in numerically predicted and experimentally obtained deformations of the Cu FAB under various ultrasonic intensities.     

Evaluation on plastic deformation property of copper nano-film by nano-scale cantilever specimen

We investigate the elasto-plastic deformation properties of a 20-nm-thick copper (Cu) thin film. A nano-scale cantilever specimen is fabricated from multilayer thin films, where the Cu thin film is sandwiched between a silicon nitride layer and a silicon substrate. During bending, the load, P, and displacement, d, are carefully monitored using an electron microscope, and a distinct non-linearity is observed. The plastic constitutive equation of the Cu thin film, which is assumed to obey a power hardening law (σ = Rεⁿ (σ > σ_y)), is inversely derived by finite element method fitting the experimental results. The residual stress in each layer is experimentally examined, and the effect is included in the inverse analysis. We obtain σ = 3316ε^0.29 [MPa] and a yield stress of 765 MPa for the Cu film. The yield stress is about 10 times higher than that of the bulk, and the exponent is also larger. Moreover, inverse analysis based on the bending experiment data, without considering the residual stress, gives a good approximation of the plastic law. This is because the plastic deformation preferentially takes place at the top and bottom surfaces, where the residual stress is relieved during fabrication of the specimen.     

Experimental analysis and finite element modelling of nano-scratch test applied on 40-120 nm SiCN thin films deposited on Cu/Si substrate

In this work, the nano-scratch test is used to characterize the interfacial adhesion of amorphous SiCN thin films deposited by plasma enhanced chemical vapour deposition on Cu/Si substrates. The experimental results show that the critical load F_c is directly related to the rupture of the SiCN/Cu interface. A strong linear dependence of F_c to the SiCN thickness independently to the adhesion is also put in evidence. A three-dimensional finite element model of the test is then built. The results show a clear relation between the stresses into the coating and the cracking and buckling of the film observed experimentally. We then discuss how the interfacial tensile stresses can explain the increase of F_c with the film thickness.     

Nanomechanical characterization of thermally evaporated Cr thin films — FE analysis of the substrate effect

We study the substrate effect on the deformation and hardness behaviour of chromium thin films using nanoindentation technique. Two different substrates namely Si (100) and AISI-304 SS are used in order to obtain a soft film on a hard substrate and a hard film on a soft substrate combination. Typical hardness variations for the two combinations are obtained. It is also observed that Cr thin films deposited on two different substrates deform distinctly. Radial cracks are found to develop in the case of Cr film on Si whereas circumferential cracks are produced in the case of Cr film on SS substrate. Using 2-D finite element analysis, it is found that the substrate not only affects the development of plastic zone but also the stress distribution in the films which results in observed distinct hardness and deformation behaviour.     

Toughening due to domain switching in single crystal ferroelectric materials

In this paper Mode I steady state crack growth in single crystal ferroelectric materials is investigated. Specifically, the fracture toughness enhancement due to domain switching near a steadily growing crack tip is analyzed. For this purpose, an incremental phenomenological constitutive law for single crystal ferroelectric materials is implemented within a finite element model to calculate the stress and remanent strain fields around the crack tip. Also, the ratio of the far field applied energy release rate to the crack tip energy release rate, i.e. the toughening, is calculated. The numerical computations are carried out for single crystal ferroelectric materials of tetragonal or rhombohedral structure with different switching hardening and irreversible remanent strain levels. Toughening levels for crack growth along different crystallographic directions and planes are obtained and compared. Results from numerical computations for the toughening anisotropy for both tetragonal and rhombohedral crystals are presented and discussed.