Expanding Cavity Model Research Articles

New expanding cavity model for conical indentation and its application to determine an intrinsic length scale of polymeric materials

New expanding cavity model for conical indentation and its application to determine an intrinsic length scale of polymeric materials

Raman Characterization of the In-Plane Stress Tensor of Gallium Nitride

Experimental characterization of the in-plane stress tensor is a basic requirement for the development of GaN strain engineering. In this work, a theoretical model of stress characterization for GaN using polarized micro-Raman spectroscopy was developed based on elasticity theory and lattice dynamics. Compared with other works, the presented model can give the quantitative relationship between all components of the in-plane stress tensor and the measured Raman shift. The model was verified by a calibration experiment under step-by-step uniaxial compression. By combining the stress characterization model with the expanding cavity model, the in-plane residual stress component field around Berkovich indentation on the (0001) plane GaN was achieved. The experimental results show that the distributions of the stress components, which significantly differed from the distribution of the Raman shift, were closely related to the GaN crystal structure and exhibited a gradient along each crystal direction.

An incremental indentation energy method in predicting uniaxial tensile properties of ferritic-austenitic dissimilar metal welds from spherical indentation tests

Based on the expanding cavity model and energy analysis, an analytical derivation was first conducted to develop a correlation between the increment of elastic and plastic portions of external work and the von Mises stress-strain increment, yielding an incremental indentation energy method (IIEM) to determine the stress-strain relationship from spherical indentation tests (SITs). The IIEM has two potential approaches in determining the proportional limit, one from indentation plastic zone radius rp measurements and the other from trial-and-error. Through an application of digital image correlation on the various zones of ferritic (SA508Gr.3Cl.1) to austenitic stainless steel (316L) dissimilar metal welds (DMWs), it was found that the rp measurements is capable for the two bases, but failed in the other six zones due to the inhomogeneous characteristic of DMWs. In this case, the mean strain development of indentation plastic zone was extensively investigated through systematic finite element analysis to establish an IIEM independent of rp measurements. Solidarity of the IIEM was verified through experiments on the eight representative zones of DMWs, and compared with its previous counterpart (the empirically concluded Kwon method). It was found that the IIEM can restrain errors caused by the compensation effect in reverse predictions, leading to more accurate strength predictions (with a maximum error 9.85%). This study is expected to provide a convenient and reliable approach for predicting the uniaxial tensile properties from SITs, especially when the rp measurement is not applicable.

Stress distribution model for the entire plastic zone of conical indentation with power-law hardening materials

<p indent="0mm">The classical contact mechanics provides an elastic conical indentation solution, and verifies that the strain field of elastic-plastic indentation with an arbitrary blunt indenter mostly satisfies the expanding cavity model. This study reviews the stress solution of the elastic zone for conical indentation using power-law hardening materials derived from predecessors based on the expanding cavity model and elastic-plastic mechanics. Furthermore, the expression of von-Mises equivalent stress in the plastic zone of conical indentation is determined according to the plastic stress solution of an internally pressurized thick spherical shell. The numerical verification revealed that the calculated equivalent stresses via finite element analysis are consistent with the theoretical solution of the elastic zone for conical indentation. However, a clear deviation in the entire plastic zone for the power-law hardening materials is still observed. Furthermore, to accurately describe the equivalent stress in the entire plastic zone of conical indentation for power-law hardening materials, a novel model for the indented stress distribution is established by combining the dimensional analysis method and finite element simulation. Finally, the stress model was verified using a wide range of ductile materials. The comparison results revealed that all of the calculated equivalent stresses are consistent with the model predictions for the entire plastic zone.

Modeling of strengthening mechanisms of surface layers in burnishing process

In the present work, a physic-based analytical model has been developed to find the hardening mechanism contributed to surface property enhancement during burnishing process. The model takes into account the effects of grain size evolution, phase change and plasticity-induced twinning. Here, the deformation parameters were firstly modeled using the expanding cavity model and theory of incremental plasticity. Then, the strengthening mechanism are identified based on developed physics based material model. Series of experiments were carried out to confirm the hardness values obtained from the model. Finally, the developed multiphysics model were utilized to identify the influence of burnishing parameters on hardness distribution and contributed mechanisms. The obtained results indicated that the there is good agreement between the measured and predicted values of hardness. On the other hand, it was found that the twinning-induced plasticity followed by phase change (from austenite to ferrite) have more dominant influence on hardening compared to grain size evolution; however, the latter only affects the hardening up to limited depth. In addition, the burnishing depth has been identified as most influential parameter that affects hardness and hardened depth.

Theoretical study including physic-based material model to identify underlying effect of vibration amplitude on residual stress distribution of ultrasonic burnishing process

The potential of distribution of compressive residual stress at deep surface layers caused by ultrasonic-based surface sever plastic deformation (SSPD) has attracted researchers' attention to deeply study the mechanism of the process by simulation models. In majority of the researches, macro-mechanical Johnson-Cook (JC) model was utilized to govern material constitutive behavior. However, the large differences between measured and predicted values of residual stress of some materials imply that the underlying mechanism is still unclear and merits further studies. In the present work, a new physics-based constitutive equation incorporating micromechanics of initial grain size, dislocation evolution, dislocation drag and acoustic softening has been developed to obtain plastic flow stress. Then, the mechanic of the ultrasonic burnishing process incorporating superposition of static and dynamic loads was simulated using concept of expanding cavity model (ECM). The residual stress has been further calculated through combination of developed models aiming to discover the undying mechanism of vibration amplitude effect on stress fields. In order to confirm the obtained results, series of ultrasonic burnishing experiments have been carried out on AA6061-T6 under different vibration amplitude. The obtained results revealed that the developed analytical models of residual stress field are considerably more accurate than those previously established that took the macro-mechanical constitutive models to calculate plastic flow stress. Through the developed simulation model, it was studied in depth that how the vibration amplitude as the most adjustable ultrasonic vibration parameter determine variation of residual stress field. It has been demonstrated that the acoustic softening because of presence of ultrasonic energy field has the greatest impact in generation of compressive residual stress in surface and beneath layers.

Improving the tensile property calculations with plastic zone radius measurements in depth-sensing spherical indentation tests

Depth-sensing spherical indentation tests (SITs) have been widely used in tensile property calculations, but the accuracy and reproducibility of calculations may be significantly influenced by displacement measurement errors. Taking two representative tensile property calculation methods as examples, namely the analytical and numerical methods, the rationale as to why accurate and reproducible tensile property calculations cannot be expected from the depth-sensing SITs was discussed in detail. Subsequently, the proportional limit σ0 calculation from plastic zone radius rp measurements, which was analytically developed in the expanding cavity model (ECM) and experimentally measured by digital image correlation (DIC), was introduced to enhance the accuracy and reproducibility of the two representative methods. Principles for setting the strain threshold εth were established, and factors influencing the σ0 calculation from rp measurements were investigated through the optical system, the friction condition, the hardening behaviors of specimen materials, and the indentation depth. Through finite element calculations, it was proven that tensile property calculations at the existence of displacement measurement errors, particularly the constant error from the origin correction, can be significantly improved with the introduction of rp measurements. Similar findings were also observed in experiments on four metals that exhibited different hardening behaviors.

Revealing the size-independent hardness for geometrical similar indentations in metal

Based on the expanding cavity model, a new expression of hardness has been proposed, which suggests that hardness is determined by the ratio of the iso-stress surface size r to the indentation depth h. In combination with the experiment results and the finite element modeling analysis, we proved that the size-independent hardness for geometrical similar indenter is fundamentally due to the similarity of the stress distribution during the indentation process. This study first reveals the cause of size-independent hardness for geometrical similar indentation through experiments and theoretical analysis without considering the complex constitutive models.

A 3D Finite Element Model of Rolling Contact Fatigue for Evolved Material Response and Residual Stress Estimation

Rolling bearing elements develop structural changes during rolling contact fatigue (RCF) along with the non-proportional stress histories, evolved residual stresses and extensive work hardening. Considerable work has been reported in the past few decades to model bearing material hardening response under RCF; however, they are mainly based on torsion testing or uniaxial compression testing data. An effort has been made here to model the RCF loading on a standard AISI 52100 bearing steel with the help of a 3D Finite Element Model (FEM) which employs a semi-empirical approach to mimic the material hardening response evolved during cyclic loadings. Standard bearing balls were tested in a rotary tribometer where pure rolling cycles were simulated in a 4-ball configuration. The localised material properties were derived from post-experimental subsurface analysis with the help of nanoindentation in conjunction with the expanding cavity model. These constitutive properties were used as input cyclic hardening parameters for FEM. Simulation results have revealed that the simplistic power-law hardening model based on monotonic compression test underpredicts the residual generation, whereas the semi-empirical approach employed in current study corroborated well with the experimental findings from current research work as well as literature cited. The presence of high compressive residual stresses, evolved over millions of RCF cycles, showed a significant reduction of maximum Mises stress, predicting significant improvement in fatigue life. Moreover, the predicted evolved flow stresses are comparable with the progression of subsurface structural changes and be extended to develop numerical models for microstructural alterations.Graphic

Revealing the Hardness Stability for Geometrical Similar Indentations

Based on the expanding cavity model, a new expression of hardness has been proposed, which suggests that hardness is determined by the ratio of the radius of the characteristic surface r to the depth of indentation h ( r / h ). In combination with the experiment results and the finite element modeling analysis, we proved that the stability of the hardness for geometrically similar indenter is fundamentally due to the similarity of the stress distribution during indentation process. This study first approves the existence of stable hardness for geometrical similar indentation through experiments and theoretical analysis without considering the complex constitutive models.

Effect of pressure sensitivity on spherical indentation response of shape memory alloys

Instrumented indentation has proved to be an invaluable tool to determine the small volume mechanical behaviour of shape memory alloys (SMAs). Multi-axial experiments show that SMAs can exhibit pressure sensitive martensite phase transformation, which manifests in terms of tension-compression asymmetry of the associated stress and strain. Since high magnitude of pressure can prevail during indentation, the spherical indentation response of SMAs is investigated in this work by using a constitutive model that captures the above behaviour. Finite element (FE) simulations are first performed at two temperatures (close to A s and above A f ) and corresponding to different values of pressure sensitivity parameter γ 1 for a Ni-Ti SMA. It is found that the indentation load and mean contact pressure enhance at a given depth, while the residual depth at complete unloading (for temperature below A f ) reduces with increase in γ 1. Further, the transformation zone size at a fixed load drops dramatically with increase in pressure sensitivity index. An expanding cavity model (ECM) is also developed to predict the mean contact pressure for a pressure sensitive SMA and validated against FE simulations as well as available experimental data.

Simple estimation method for strain rate sensitivity based on the difference between the indentation sizes formed by spherical-shaped impactors

The Johnson-Cook (JC) flow stress model is widely used in finite element analysis (FEA) in various fields to express the deformation behavior of metallic materials under high strain rate conditions. There is a need to facilitate an easier estimate of the JC parameter C, which is a material constant in the JC model associated with strain rate sensitivity. In this study, a simple estimation method for the JC parameter C, based on the difference between the indentation sizes formed by two spherical-shaped impactors, was proposed. The proposed method allows us to estimate the JC parameter C only by measurement of the indentation sizes formed via a high-velocity impact test, a drop impact test, or an indentation test. The fundamental equation was theoretically derived based on the energy conservation during the impact process of the impactor and the expanding cavity model, combined with the JC model and the equation of thermal conduction. To validate the proposed method, FEA was conducted using ten types of target materials under five impact velocity conditions, from 6 × 10−6 to 180 m/s. Based on the indentation depths obtained from FEA, the JC parameter C was estimated and compared with that input into FEA. As a result, it was confirmed that the estimated C is in agreement with that input into FEA within the relative errors of 10% under all impact velocity and target material conditions, except the target material of Ti6Al4V. From the results, it was demonstrated that the proposed method can easily estimate the JC parameter C, that is, the strain rate sensitivity, in the wide range of strain rate from 102 to 105 s−1.

A strain-pattern-based spherical indentation method for simultaneous uniaxial tensile residual stress and flow property determination

In this study, a strain-pattern-based method was proposed to simultaneously determine the uniaxial tensile residual stress and flow property from a single-cycle spherical indentation test. The variation of the plastic zone radius (at the specimen surface) with uniaxial tensile residual stress was analytically investigated by the expanding cavity model. The analysis proved that the circular plastic boundary will be elliptical under the action of uniaxial residual stress (with a shrunken plastic zone radius along the loading direction and an extended plastic zone radius vertical to the loading direction), and this difference can be used to calibrate the magnitude of the residual stress. The analytical result was verified and modified through finite element calculations, after which a set of regression functions for Holloman hardening metals was established for load compensation, proportional limit correction, and hardening exponent calculation. The effectiveness of the method was verified through finite element calculations of spherical indentation tests on 16 Holloman hardening and 6 metals used in engineering applications at different residual stress levels. The verification proved that maximum errors for strength and residual stress calculations are about 10% and 15%, respectively, and the potential of the new proposed method was validated.

Influence of SD effect on Johnson–Cook hardening constitutive material model: Numerical and experimental investigation for armor steel

The influence of strength–differential (SD) effect has been investigated on Johnson–Cook (JC) material model parameters for armor grade RHA steel. Two set of JC model parameters has been developed with the help of compressive and tensile flow curve. Further, a finite element analysis (FEA) of static (at 10−3 s−1) and dynamic indentation (at 104 s−1) has been performed to validate the effect of tensile and compressive JC material model parameters. In addition, static and dynamic indentation FEA results has been compared with the expanding cavity model (ECM) and experimental Split–Hopkinson pressure bar results, respectively. FEA simulation with compressive data has been shown good agreement with experimental results and analytical models. %SD effect is increased with an increase in plastic strain in armor grade RHA steel.

Compressive Properties of Nanoporous Gold Through Nanoindentation: An Analytical Approach Based on the Expanding Cavity Model

We investigated the analytic relation between hardness and compressive yield stress using an expanding cavity model (ECM) for nanoporous gold (np-Au). We prepared three np-Au samples with ligament sizes 30.61, 59.36 and 116.33 nm by free-corrosion dealloying and post heat treatment. The indentation contact morphology was examined to estimate the hardness accurately from the nanoindentation load-depth curve. Unlike conventional dense metals, the deformation was confined to the projected contact area, and the center of the residual impression was dominated by densification. The projected contact area estimated by the Oliver–Pharr method was overestimated, so that a new contact area function was proposed that considered the indentation contact morphology of np-Au. It was confirmed that a hardness value taking into account the indentation contact morphology of np-Au matches well with the hardness derived by direct measurement of the residual impression. We modeled the ratio of hardness to compressive yield stress for np-Au using an ECM. The scaling factors, which represent the extra strain-hardening in the core in the ECM, were analyzed for np-Au and dense metals. An ECM that better matches np-Au is suggested based on the scaling factor resulting from densification beneath the indenter.

Characterization of elastic-plastic properties of surface-modified layers introduced by carburizing

The mechanical performance of surface-modified layers is important during mechanical-component service. A simpler expression to characterize the carbon content of the surface-modified layers was proposed for alloy steel after carburizing. The tensile test method was used to determine the elastic–plastic properties of the surface-modified layers of the 18CrNiMo7-6 carburized alloy. The hardness of all tensile test specimens was determined and the relationship with the carbon content was given. The elastic modulus, yield strength and hardening exponents were obtained for the surface-modified layer from the tensile test and stress–strain curves. The elastic modulus does not change with depth, whereas the yield strength and hardening exponent decreased gradually along the depth and were almost correlated with the carbon distribution. The hardening exponent model of the surface-modified layer was given as an exponential function. The relationship between elastic modulus, yield strength, hardening exponent and hardness agrees broadly with the modified expanding cavity model for the surface-modified layer of 18CrNiMo7-6 carburized alloy.

Indentation densification of fused silica assessed by raman spectroscopy and constitutive finite element analysis

AbstractInelastic deformation of anomalous glasses manifests in shear flow and densification of the glass network; the deformation behavior during indentation testing is linked strongly to both processes. In this paper, the indentation densification field of fused silica is investigated using depth‐resolved Raman spectroscopy and finite element simulations. Through affecting the size of the indent, the normal load and the Raman laser spot size determine the spatial sampling resolution, leading to a certain degree of structural averaging. For appropriate combinations of normal load (indent size) and laser spot diameter, a maximum densification of 18.4% was found at the indent center. The indentation behavior was modeled by extended Drucker‐Prager‐Cap (DPC) plasticity, assuming a sigmoidal hardening behavior of fused silica with a densification saturation of 21%. This procedure significantly improved the reproduction of the experimental densification field, yielding a maximum densification of 18.2% directly below the indenter tip. The degree of densification was found to be strongly linked to the hydrostatic pressure limit below the indenter in accordance to Johnson's expanding cavity model (J. Mech. Phys. Solids, 18 (1970) 115). Based on the good overlap between FEA and Raman, an alternative way to extract the empirical correlation factor m, which scales structural densification to Raman spectroscopic observations, is obtained. This approach does not require the use of intensive hydrostatic compaction experiments.

Evaluation of tensile yield strength of high-density polyethylene in flat-ended cylindrical indentation: An analytic approach based on the expanding cavity model

Abstract

Expanding Cavity Model Combined With Johnson–Cook Constitutive Equation for the Dynamic Indentation Problem

Abstract The indentation formed on a metallic component by the high-velocity impingement of a small object can fracture the component, and this is known as foreign object damage. In this type of dynamic indentation, it is necessary to consider the effects of work hardening, strain rate hardening, and thermal softening in the impinged material. In this study, in order to consider these effects, the expanding cavity model based on a spherical formulation is modified via the Johnson–Cook constitutive equation for the dynamic indentation problem. Additionally, an equation is developed based on energy conservation and the modified expanding cavity model to predict the size of the indentation formed by an impingement of a solid sphere (EPIS). The distributions of equivalent plastic strain, equivalent plastic strain rate, temperature, and equivalent von Mises stress obtained via the expanding cavity model were in good agreement with the data obtained from the finite element analysis (FEA). Furthermore, it was demonstrated that EPIS accurately predicted the indentation size formed on various metallic materials at several impingement velocities in the range of 50–300 m/s. Consequently, EPIS can be effectively applied to an impingement problem of a hard sphere onto a sufficiently thick ductile material within 300 m/s without any help of FEA.

Prediction of Indentation Size Formed by a High-Velocity Impingement of a Solid Sphere Based on an Expanding Cavity Model

The fan blades and turbine blades in a jet engine are seriously damaged by high velocity impingements of various foreign objects. In this study, a prediction method of indentation size formed by a high-velocity impingement of a solid sphere (PMIS) was developed from a theoretical model based on an expanding cavity model and energy conservation before and after impingement. The Johnson-Cook constitutive equation was employed to introduce effects of work hardening, strain rate hardening and thermal softening into the cavity model. As a result, the distribution of equivalent plastic strain, equivalent plastic strain rate, temperature and equivalent von Mises stress estimated using the expanding cavity model was in good agreement with the data obtained from the finite element analysis. In addition, it has been demonstrated that PMIS can accurately predict the radius of indentation formed on various metallic materials subjected to the impingement of a solid sphere with the radii of 0.75, 1.5 and 3 mm at several impact velocities from 50 to 300 m/s.