Load-depth Curves Research Articles

Stress–Strain Curve Estimation from Load–depth Curve of Spherical Indentation Test Based on Finite Element Analysis and Optimization

Stress–Strain Curve Estimation from Load–depth Curve of Spherical Indentation Test Based on Finite Element Analysis and Optimization

Determining the bilinear flow stress of thin films by comparing load/depth and sink-in/pile-up behavior between nanoindentation and FEM

Abstract It is known from the literature that the load-depth curve obtained by nanoindentation cannot uniquely determine the plastic flow stress behaviour of the material. In this study, the sink-in/pile-up of nanoindentation is another important information for predicting the elastic-plastic behaviour of thin film. Therefore, by comparing load/depth behaviour and sink-in/pile-up from nanoindentation with finite element analysis, the bilinear flow stress of the coating can be determined. Commercial TiN coatings are currently used as an example for this study. The bilinear flow stress of TiN can be determined by this method: the elastic modulus of 343.007 GPa, the yield stress of 10.458 GPa and the tangent modulus of 214.080 GPa. In addition, the stress of loading and unloading during the nanoindentation of the thin film was also investigated.

Study on the atomic scale deformation mechanism of the h-BN coating nickel matrix composites

In this paper, molecular dynamics (MD) simulations are applied to study the contact mechanical behavior of hexagonal boron nitride (h-BN) coating/Ni matrix composites during nanoindentation, revealing the material deformation mechanism. A systematic study is carried out on the load-depth curve, temperature response curve, defect morphology, and dislocation density in indentation. The results show that the presence of h-BN coating increases the actual loading area of the matrix, effectively reduces the contact stress between the tip of the indenter and the specimen, and also induces the generation of high-density Shockley dislocations, which increases the dislocation entanglement and leads to the localized hardening of the composites, thus reducing the matrix damage. During the loading process, the temperature of the specimen before the destruction of the h-BN coating is significantly lower than that after the destruction, indicating that the coating tends to block the transfer of temperature to the interior of the specimen, preventing the occurrence of temperature-induced tissue phase changes inside the specimen, and providing good protection for the interior of the specimen.

Evaluation method of dynamic indentation behavior of glass based on electromagnetic induction phenomena

AbstractContact damage of glass is one of the most crucial issues for glass products. To develop strong and tough glass products and to compare damage resistance among glass compositions, a simple method for evaluating the mechanical response of glass during contact is required not only for glass mechanists but also for glass customers and suppliers. Although it is well known that the quasi‐static Vickers indentation test is one of the simplest and most useful methods to evaluate hardness and brittleness in glass, the indentation response of glass under the indenter at higher impact velocities remains to be quantitively understood because of the difficulty of measurement and limited experimental works. In this study, therefore, the dynamic indentation behavior of soda‐lime glass is evaluated by using a lab‐made free‐drop indentation set‐up with the coils for detecting electromotive forces (EMFs). The cono‐spherical indenter made of tungsten carbide attached with a neodymium magnet was employed to generate the EMFs when the indenter passed through the coils located near the glass sample. The impact load versus indentation depth curve during the impact within a few tens of microseconds was successfully obtained both for an elastic contact and for an inelastic contact. Under an elastic condition, where no residual indent nor any cracks were left on the glass surface after the test, it is confirmed that there is almost no hysteresis in the impact load versus indentation depth curve and that the curve can be reproduced by the Hertzian analytical solution. Under an inelastic condition, on the other hand, it is found that the hysteresis in the impact load versus indentation depth curve stems from inelastic phenomena, such as plastic deformation (shear flow and/or permanent densification) and cracking. These results suggest that the dynamic indentation technique based on electromagnetic induction phenomena is a useful and effective tool for evaluating the mechanical responses of glasses during the impact.

Investigation of in-plane visco-deformation in PMMA indentation via TLV model calibrated with tensile specimens

Time-dependent in-plane deformation in PMMA indentation is investigated via two-layer visco-plasticity model (TLV). The parameters of TLV model for PMMA were calibrated for tensile specimens using finite element analysis (FEA) and an optimization approach. The TLV model with the calibrated parameters was then employed to simulate nano- and micro-indentation behaviors including creep, relaxation, and other responses under various test conditions. Comparisons of experimental and FEA indentation load-depth curves showed agreement except the last unloading segment. This discrepancy was attributed to the inability of TLV model in replicating the delayed elastic recovery. In addition, deformation recovery with in-plane displacement (ui) was observed over a few days after indentation load removal. The characteristics of ui recovery and the factors influencing its magnitude are discussed. The proportionality of relative ui recovery rate prominently featured in the viscoelastic properties of PMMA, regardless of indentation history.

Identification of Elastoplastic Constitutive Model of GaN Thin Films Using Instrumented Nanoindentation and Machine Learning Technique

This study presents a novel inverse identification approach to determine the elastoplastic parameters of a 2 µm thick GaN semiconductor thin film deposited on a sapphire substrate. This approach combines instrumented nanoindentation with finite element (FE) simulations and an artificial neural network (ANN) model. Experimental load–depth curves were obtained using a Berkovich indenter. To generate a comprehensive database for the inverse analysis, FE models were constructed to simulate load–depth responses across a wide range of GaN thin film properties. The accuracy of both 2D and 3D simulations was compared to select the optimal model for database generation. The Box–Behnken design-based data sampling method was used to define the number of simulations and input variables for the FE models. The ANN technique was then employed to establish the complex mapping between the simulated load–depth curves (input) and the corresponding stress–strain curve (output). The generated database was used to train and test the ANN model. Then, the learned ANN model was used to achieve high accuracy in identifying the stress–strain curve of the GaN thin film from the experimental load–depth data. This work demonstrates the successful application of an inverse analysis framework, combining experimental nanoindentation tests, FE modeling, and an ANN model, for the characterization of the elastoplastic behavior of GaN thin films.

A dual indentation method for evaluating intrinsic material properties of metals under residual stress

A dual flat-spherical indentation (DFSI) method is proposed for evaluating elastic-plastic properties of metallic materials under equi-biaxial residual stress (RS) state. DFSI consists of two indentation steps: (step I) performing flat indentation (FI) on a specimen having significant presence of residual stress, and (step II) performing spherical indentation (SI) at the center of residual imprint generated by the FI (step I). Indentation load-depth curve in conventional single spherical indentation (SSI) generally varies with RS, while the load-depth curves of SI step in DFSI are almost identical even under significant RS; this is attributed to stress redistribution during the preceding FI step. A set of FE analyses are performed for a wide range of properties of general metals. Regression-based mapping functions are established for evaluating the elastic-plastic properties from the SI data of step II. Finally, the proposed method is validated by performing DFSI experiments on the mechanically stressed SM45C and SS400 specimens.

An Investigation of the Effect of Wide Range Gamma Radiation from Nanoindentation of the SAC305 Solder Alloy

This study utilises nanoindentation testing to investigate the impact of varying gamma radiation doses on the micromechanical properties of Sn-Ag-Cu (SAC) alloy. Specifically, the focus is on evaluating changes in hardness, reduced modulus, and creep behaviour. The stencil-printed method and reflow soldering process were employed to apply the SAC solder paste and create solder joints on the surface of the printed circuit board. The soldered samples underwent exposure to gamma radiation at different doses, specifically 5, 50, 500, 5000, and 50000 Gy. The solder received in its original state was used as the control sample. Subsequently, the samples were subjected to a nanoindentation test in order to ascertain the correlation between load and depth, depth and dwell time, when exposed to radiation. The load-depth curve results indicate that there is a transition in the behaviour of solder joint materials from elastic to plastic deformation as the radiation dose increases. A finding has been made indicating that exposure to gamma radiation has the potential to induce a transition in the behaviour of SAC from an elastic state to a plastic state. The exposure to radiation doses has been found to induce changes in the atomic arrangement and structural properties of materials, leading to an increase in their hardness values. Nevertheless, it was observed that with increasing radiation doses up to 500 Gy, there was a noticeable decrease in the hardness value, which can be attributed to the occurrence of softening behaviour. Exposure to a high dose exceeding 5000 Gy leads to atomic displacement and transmutation products, subsequently resulting in plastic deformation. The stress exponent value signifies the occurrence of the deformation mechanism in solder material when exposed to gamma radiation. The study revealed that there was a shift in the deformation mechanism from grain boundary sliding to dislocation climb as the radiation dose increased from low to high levels.

Meso-mechanics of packed C-S-H colloids by nanoindentation: A coarse-grained molecular dynamics study

Colloidal calcium silicate hydrate (C-S-H) gel significantly contributes to cement paste's strength and durability. In this study, the coarse-grained (CG) models for packed C-S-H colloidal particles with different packing densities were established and meso-mechanically assessed via nanoindentation. Load-depth curves showed indentation hardness values (0.55, 1.16, and 2.63 GPa) for the systems with packing densities (η) of 0.50, 0.55, and 0.60, respectively. Structurally, the nano-indenter had a broader impact on low-density C-S-H (η=0.5, impact radius = 118 and 150 nm respectively at indentation depths = 50 and 100 nm) than high-density C-S-H (η=0.6, impact radius = 106 and 140 nm at the same depths). Low-density colloids were easily compressed without deforming low-depth nearby regions, while high-density colloids were squeezed laterally, causing deformation in these regions. Packed C-S-H colloids displayed two-stage stress relaxation behavior: rapid initial relaxation due to nanoindentation-induced instability, followed by slower relaxation due to C-S-H's viscous nature. Furthermore, higher loading rates caused initial unstable deformation, but better stability after stress relaxation compared to lower loading cases. However, the effect of loading rate on the impact region was negligible. These meso-level insights enhance our comprehension of the C-S-H gel properties in cement paste as well as the nanoindentation mechanics.

Elastic-plastic properties of ceramic matrix composites characterization by nanoindentation testing coupled with computer modeling

The nanoindentation test is utilized to measure the hardness and elastic modulus of various components within Ceramic Matrix Composites (CMCs), including continuous stiffness measurement (CSM) and quasi-static testing. To further characterize the elastic-plastic properties of CMCs, an efficient method is proposed by using backpropagation (BP) neural network inversion, which is based on the relationship between load and depth obtained by the nanoindentation experiment and Abaqus finite element simulation. As part of the analysis, the load-depth (p-h) curve is simplified to six characteristic parameters through the fitting of the experimental results. The pre-estimated elastic-plastic characteristics are utilized to model the nanoindentation test for CMCs. A BP neural network is used to perform deep learning on simulation data, which leads to a network structure that matches the trained parameters of the p-h curve and elastic-plastic properties. The experimental data for fiber is inputted into the network to acquire the elastic-plastic properties required in practice. The accuracy of the proposed method is verified by comparing the inversion results with the simulation outcomes.

Impacts of Abnormal Nanoindentation Points on Micromechanical Properties and Content of Phase in Hydrated Cement Paste

During nanoindentation analysis of cementitious materials, data points with abnormal load-depth curves are often obtained. This study investigates the effect of these abnormal indentation points (AIP) on the micromechanical properties and content of phase in hydrated cement paste calculated by nanoindentation test combined with deconvolution analysis, including the least-square-estimation (LSE) and maximum-likelihood-estimation (MLE) methods. The results indicate that the AIP is mostly associated with phases with low mechanical properties, and the exclusion of AIP significantly affects the volume fractions of micropore phase and low-density calcium silicate hydrate, while the mechanical properties of phases keep stable except that the mechanical properties of micropore phase are slightly increased by the exclusion of the AIP. The phase contents derived based on LSE showed a more significant change than those derived based on the MLE when AIP was excluded from indentation data. In addition, the phase content derived by nanoindentation analysis was compared to that derived by other analysis methods, including mercury intrusion and quantitative x-ray diffraction.

Physics-informed neural networks for spherical indentation problems

A scientific deep learning (SciDL) approach was developed by integrating a regression-based spherical indentation method with an artificial neural network (ANN) to extract elastic–plastic properties from indentation load-depth curves. Different combinations of material parameters are constructed through Latin Hypercube Sampling (LHS) process to create a database of indentation parameters. An attempt is made to reversely obtain load-depth (P-h) data using a regression function for a given set of material parameters; this method is further verified by performing finite element (FE) simulations. SciDL models i.e., physics-informed artificial neural network (PI-ANN) with autoencoder (AE) are built based on PyTorch library, and the models are trained using the generated database. Transfer learning (TL) techniques are employed to achieve better training performance with the PI-ANN model. Compared with data-driven models, SciDL models produce consistent predictions with higher accuracy; the coefficient of determination R2 values are observed greater than 0.960. TL techniques allows the SciDL model to learn much faster (≈ 42 epochs) than traditional method (≈ 2400 epochs). Finally, we perform spherical indentation experiments on STS304 and SM45C, and validate the performance of the trained SciDL models; AE integrated PI-ANN model with tanh activation function predicts the material properties close to reference values of SS400 and SM45C. The proposed SciDL approach can be extended for characterizing engineering materials and structures by incorporating any priorly developed mechanical testing method with ANN.

Exploring the Nb2O5 coating deposited on the Ti-6Al-4V alloy by a novel GE-XANES technique and nanoindentation load-depth

This research group has been demonstrating the significant advantages of using Nb2O5 coatings for functionalizing titanium, aluminium, and stainless steels. Regarding the biomedical sector and considering Ti-6Al-4V alloy, the reactive sputtering technique improved the cell viability, the osteogenic performance of cells involved in the osseointegration process as well as the ability to delay bacterial proliferation. The characteristics of the Nb2O5 coatings were assessed before by using standard methods, which provide information only a few tens of nanometers depth. Given that the Nb2O5 coating fabricated in this work exhibits a thickness of approximately 300 nm, the GE-XANES technique emerges as the most suitable method for this analysis. Additional information was provided with the aid of nanoindentation load-depth (P-h) curves. GE-XANES results indicated the formation of a homogeneous layer of Nb2O5 coating on the Ti-6Al-4V surfaces. The deposition process improved the surface hardness of the Ti-6Al-4V alloy (4.38 GPa versus 5.62 GPa) considering the 2 mN load.

Surface hardness and scratch characteristics of nanosecond laser colored Ti-based metallic glass

Pulsed laser irradiation is widely used for coloring metallic glass (MG) surfaces due to its efficient and clean processing characteristics. However, the hardness and tribological properties of laser colored MG surfaces have rarely been studied. In this study, various colors were prepared on Ti-based MGs by nanosecond pulsed laser irradiation, and the mechanical properties of the colored surfaces were subsequently investigated. Before hardness testing, the colored surfaces were slightly polished and characterized for surface morphology and phase composition. The hardness of the colored surfaces was measured by a nanoindentation instrument, and the corresponding load depth curves and residual indentations were obtained. The results showed a 39% increase in hardness compared to the as-cast surface. Scratch tests were performed to investigate the tribological properties of the colored surfaces. The residual scratches and three dimensional (3D) topographies exhibited diverse features, and the scratch damage mechanism was analyzed accordingly. This study broadened the knowledge of surface hardness and tribological properties, and illustrated the application potentials of laser colored MGs.

Parameter determination of anisotropic yield function using neural network-based indentation plastometry

Understanding the impact of plastic anisotropy on the formability of sheet metals is crucial for their industrial application and high-precision forming simulation. Existing multiple uniaxial tensile tests for measuring the plastic flow of anisotropic materials, however, are costly, time-consuming, and destructive. Therefore, it is more efficient to focus on indentation plastometry, a simple, nondestructive test that can quickly extract tensile properties. The objective of this work is to directly derive the parameters of the advanced Poly6 yield criterion and hardening, which describes strong plastic anisotropy, using indentation plastometry basd on a neural network (NN) system. The identification process for these parameters through conventional tensile tests is inherently complex, thereby determining the parameters directly from indentation data presents an unprecedented challenge. We trained NNs using a database generated from verified finite element (FE) simulations of spherical indentations. To systematically iterate these FE simulations, we designed a strategy generating a set of input anisotropic parameters that ensure the convexity of the yield function. We considered the radial and vertical displacement fields around the indentation mark along with the load-depth curve as indentation responses. Through a comprehensive analysis of the correlation between displacement profiles, we have proposed an optimal feature extraction method for NN training. The developed FE-NN model was evaluated by comparing the predicted parameters from the indentation responses of the target materials with those measured from tensile tests. These parameters were expressed as the yield locus and directional properties. The results demonstrated that the FE-NN modeling approach is robust and can accurately capture the anisotropic plastic flow from indentation responses.

An improved analytic model for extracting plastic mechanical property of matrix in hot-extruded in-situ TiB2/2024 composite based on Berkovich indentation experimental results

Due to the strengthening mechanisms mainly contributed to dispersed nano TiB2 particles, plastic mechanical property of Al matrix in hot-extruded in-situ TiB2/2024 composite, referring to yield strength (YS) and strain hardening exponent (SHE), are considerably improved compared with 2024 Al alloy. In addition, the plastic mechanical property of Al matrix cannot be obtained directly through traditional uniaxial tensile tests. After carrying out dimensional analysis and FE simulations, an improved analytical model with no need of characteristic stress/strain is proposed found on existing model, to extract plastic mechanical property parameters of Al matrix based on sufficient Berkovich indentation experimental results. The indentation experiments carried out using the custom method “G-Series Basic Hardness, Modulus, Tip Cal, Load Control” were implemented in case of four different maximum indentation loads to get indentation load-depth curves (ILDCs) of Al matrix, where the corresponding morphologies of residual indents were obtained by SEM observation. With these experimental results as input, the YS and SHE of Al matrix can be calculated by utilizing the proposed analytic model. To verify the effectiveness of proposed improved model, a unit-cell FE model with same particle volume fraction as the macroscopic composite is established and analyzed, and the calculated results are in good agreement with the uniaxial tensile test results.

Novel procedure to determine the stress-strain relation of Lithium-ion Batteries through indentation test

This study proposes a novel approach to convert the load–depth curve measured in the experimental indentation test into stress–strain curves. Due to the lack of the tensile test results for the cylindrical lithium-ion battery (LIB) cell, a combination of the analytical analysis and the inverse optimization approach is first conducted as a reference point to determine the elastoplastic constitutive equation, which can predict the response of cylindrical LIB cell cells under the cylindrical indenter. The obtained load-depth curves from the robust optimization algorithm are in good agreement with the experimental loading–unloading curves. This novel approach includes a new definition of the contact area between the cylindrical indenter and the cylindrical LIB cell. Finite element models validate the measured contact area values from the analytical approach. The analytical approach in this study can accurately capture the stress–strain curve validated with optimization results. Afterward, different single indentation tests are conducted to estimate the reduction of Young’s modulus through the increasing indentation depth. Noteworthy, a definition of the indentation strain can physically interpret the indentation strain in the case of two crossed cylinders. The cell voltage is measured during the indentation test to detect the internal short circuit in the LIB cell. In homogenized modeling, the damage of the LIB cell is considered, which is an essential indicator of the internal short circuit of a LIB cell and possible thermal runaway.

Computational modeling and mechanical properties of 3Y-TZP ceramic composites reinforced by mullite whiskers with different diameters

An accurate characterization of microstructure is essential to realize the performance prediction and design optimization of the ceramic matrix composites (CMCs). In this work, according to the feature of in-situ growth of mullite whiskers in 3 mol% Y2O3 stabilized tetragonal ZrO2 (3Y-TZP) ceramics, 2 vol% whiskers of different diameters formed in the 3Y-TZP ceramics were numerically reconstructed based on Monte Carlo method to explore the mechanical properties of the ceramic composites. The results show that the nanoindentation model of whisker reinforced 3Y-TZP ceramics can effectively predict the hardness of the composite ceramics according to load-depth curves. The stress transfer mechanism between the ceramic matrix and the whiskers can also be found. It is found that the hardness of the composite ceramics decreases gradually with the increase of the whisker diameters. According to three-point bending model, the obtained bending strength and fracture toughness of the composite ceramics increase first and then decrease with increasing whisker diameter. When the diameter of mullite whiskers is 0.12 μm, the bending strength of the composite ceramics reaches the maximum of 826 ± 25 MPa and the fracture toughness reaches the maximum of 10.5 ± 0.2 MPa•m1/2. The toughening mechanism of whisker reinforced 3Y-TZP ceramics was also demonstrated. This work provides a numerical method for the selection and optimization of parameters of reinforcement phase in preparation of 3Y-TZP ceramic composites.

Research on Determining Elastic-Plastic Constitutive Parameters of Materials from Load Depth Curves Based on Nanoindentation Technology.

It is of great significance for structural design and engineering evaluation to obtain the elastic-plastic parameters of materials. The inverse estimation of elastic-plastic parameters of materials based on nanoindentation technology has been applied in many pieces of research, but it has proved to be difficult to determine the elastic-plastic properties of materials by only using a single indentation curve. A new optimal inversion strategy based on a spherical indentation curve was proposed to obtain the elastoplastic parameters (the Young's modulus E, yield strength σy, and hardening exponent n) of materials in this study. A high-precision finite element model of indentation with a spherical indenter (radius R = 20 µm) was established, and the relationship between the three parameters and indentation response was analyzed using the design of experiment (DOE) method. The well-posed problem of inverse estimation under different maximum indentation depths (hmax1 = 0.06 R, hmax2 = 0.1 R, hmax3 = 0.2 R, hmax4 = 0.3 R) was explored based on numerical simulations. The results show that the unique solution with high accuracy can be obtained under different maximum press-in depths (the minimum error was within 0.2% and the maximum error was up to 1.5%). Next, the load-depth curves of Q355 were obtained by a cyclic loading nanoindentation experiment, and the elastic-plastic parameters of Q355 were determined by the proposed inverse-estimation strategy based on the average indentation load-depth curve. The results showed that the optimized load-depth curve was in good agreement with the experimental curve, and the optimized stress-strain curve was slightly different from the tensile test, and the obtained parameters were basically consistent with the existing research.

Effect of indenter radius on mechanical properties of B3-GaN in nanoindentation based on molecular dynamics

Molecular dynamics (MD) simulation of nanoindentation was conducted to study the indentation of sphalerite gallium nitride (B3-GaN) according to different radius spherical indenters. The effect of indenter radius on the evolution of dislocations, atomic displacements, Von Mises stresses, and phase transition of B3-GaN was primarily analyzed. The outcomes of the study indicate that the plastic deformation of B3-GaN crystals may be attributed to the propagation of dislocations and amorphization. The pop-in events of load in the load-depth curve is related to dislocation nucleation, amorphization, and phase transition. As the indenter radius increases, the critical load during the elastic-plastic transformation increases, and the hardness decreases gradually. The Burgers vector of dislocations during indentation is almost all 1/2 <110>, and the main slip directions were along the <110> crystallographic direction family. The magnitude and distribution range of atomic displacements and Von Mises stresses increase with the indenter radius, and this further promotes the nucleation and propagation of dislocations in the slip system, thus intensifying the plastic deformation of B3-GaN crystals. The elastic deformation of B3-GaN crystals may be attributed to amorphization and phase transition. During the elastic deformation phase, increasing the indenter radius promotes the phase transition of B3-GaN to fibrillated gallium nitride (B4-GaN) and enhances the amorphization of B3-GaN in the deformation layer. In addition, the increase in the indenter radius delays the generation of amorphous phases and phase transition. The outcomes of the study help us understand the mechanical properties of B3-GaN along with its potential for use in precision finishing.