Indentation Size Effect Model Research Articles

Indentation Size Effect Model of Ti6Al4V Alloy by Combining the Macroscopic Power‐Law Constitutive Relation and Strain Gradient Theory

Indentation size effect (ISE) is a critical feature for investigating the local properties of materials using the indentation test method. Although there are numerous works on the indentation response, there is a lack of data in the literature on the association between nanoindentation and uniaxial compression. Herein this study, uniaxial compression tests combined with Digital Image Correlation Technique are conducted on Ti6Al4V alloy with a focus on investigating the elastic–plastic properties and obtaining the accurately constitutive relation. Then, an ISE model of Ti6Al4V is established based on this constitutive relation and strain gradient theory. Indentation hardness, predicted by this proposed model, agrees well with the measurement by nanoindentation experimental techniques. This model can be used to bridge the gap between the uniaxial stress–strain and the depth‐dependent hardness, and clarifies the effect of the strain hardening on indentation hardness. The significance of this model is that it provides a direct and reasonable theory foundation to investigate ISE by uniaxial compression, and promotes the potential application of nanoindentation.

Berkovich nanoindentation of Zr55Cu30Al10Ni5 bulk metallic glass at a constant loading rate

Berkovich nanoindentation experiments were conducted on Zr55Cu30Al10Ni5 bulk metallic glass under a constant loading rate. Indentation hardness decreased linearly with contact depth, which defied the direct application of conventional indentation size effect model for crystalline material. Elastic modulus remained invariant under small loads, but decreased linearly under large loads. The scaling relationships between indentation variables such as contact depth, permanent depth, the maximum indentation displacement, plastic energy, the total work of indentation and their ratios were investigated. The area of shear band circle, the projected contact area at the maximum load, and the residual projected area were all found to be proportional to one another. A new methodology to determine fracture toughness was proposed based on the total work of indentation, provided that the critical indentation displacement at the onset of fracture corresponds to zero value of elastic modulus extrapolated from curve fitting.

Investigation of the mechanical and microstructural evolution of a Cu based bulk metallic glass during ion irradiation

Ion irradiation and annealing experiments were performed on Cu60Zr20Hf10Ti10 bulk metallic glass (BMG) specimens to investigate their irradiation- and temperature-induced microstructural and mechanical property evolution. For the ion irradiations, samples were exposed to 9 MeV Ni3+ ions to a midrange (~1.2 μm depth) dose of 10 displacements per atom (dpa) at temperatures ranging from room temperature to 360 °C (the corresponding peak dose at ~2.8 μm depth was ~25 dpa). Bulk X-ray diffraction (XRD) and transmission electron microscopy (TEM) characterization revealed that the alloy did not crystallize during irradiation up to 290 °C but did partially crystallize at 360 °C. XRD analysis revealed that the crystallization which occurred in the sample irradiated at 360 °C was caused by thermal effects instead of irradiation displacement damage. Subsequent Rietveld refinement analysis of the XRD measurements revealed the presence of two distinct crystal phases, namely a CuTiZr hexagonal structure belonging to the P63/mmc space group and a CuTi tetragonal structure belonging to the P4/mmm space group. Nanoindentation experiments revealed that no pronounced hardness changes occurred in the specimens irradiated at room temperature and 290°C, although significant hardening was observed in the sample irradiated at 360 °C. The significant increase in the hardness at 360°C was ascribed to thermally induced partial crystallization of the alloy instead of the ion irradiation. In general, the results of the nanoindentation experiments and XRD characterization suggest that although the Cu BMG exhibits good stability during irradiation at temperatures up to 290 °C it is not suitable for irradiation environments where the temperature is 360 °C for extended periods of time. The Lam and Chong extrapolation method, which has been used to study the indentation size effect (ISE) in amorphous alloys, was employed to quantify how irradiation and temperature affect this type of behavior in the BMG. However, the poor linear fitting of the indentation hardness data by this model indicate that a new ISE model is likely needed to quantify indentation hardening in BMGs.

Evolution of the microstructural and mechanical properties of BAM-11 bulk metallic glass during ion irradiation and annealing

Ion irradiation and annealing experiments were performed on Zr52.5Cu17.9Ni14.6Al10Ti5 “BAM-11” bulk metallic glass (BMG) specimens to evaluate their irradiation- and temperature-induced microstructural and mechanical property evolution. For the ion irradiations, samples were exposed to 9 MeV Ni3+ ions to a midrange (∼1.5 nm depth) dose of 10 displacements per atom (dpa) at temperatures ranging from 25 to 360 °C. A separate set of samples were annealed at 150 °C, 200 °C and 300 °C with respective heating times of 96, 72, and 48 h. Bulk X-ray diffraction (XRD) and transmission electron microscopy (TEM) characterization revealed that the alloy did not crystallize during irradiation up to 290 °C, although partial crystallization occurred at 360 °C throughout the unirradiated and irradiated regions of the sample. Transmission electron microscopy (TEM) characterization suggested that some of the irradiated region retained an amorphous structure, supporting the idea of partial crystallization. XRD analysis revealed that the crystallization which occurred in the sample irradiated at 360 °C was caused by thermal effects, and not irradiation displacement damage, and may be due to slight impurity contamination in the ingot. Nanoindentation experiments showed that only a slight amount of hardening was observed in the specimen irradiated at room temperature and 290 °C. However, significant hardening occurred in the sample irradiated at 360 °C (∼18% increase) as well as the unirradiated annealed specimens. For the sample irradiated at the highest temperature, the substantial increase in hardness was attributed to the partial crystallization of the alloy due to thermal, rather than irradiation effects. Overall, the results of the XRD and nanoindentation characterizations indicate good stability during irradiation at 25–290 °C but suggest that the BAM-11 BMG is not suitable for irradiation environments where temperatures exceed 300 °C for prolonged periods of time. Three different extrapolation models were employed to study how irradiation and annealing modify the indentation size effect (ISE) in the BAM-11 BMG. The poor linear fitting, as exhibited by all of the underlying equations, indicate that a new ISE model is needed to quantify nanoindentation mechanical properties in BMGs.

Indentation size effect in amorphous polymers based on shear transformation mediated plasticity

The existence of an indentation size effect (ISE) in the measured hardness in poly (methyl methacrylate) is investigated by performing nanoindentation experiments. The continuous stiffness measurement technique was employed to measure the hardness and the elastic modulus of the sample as a function of the indentation depth at different strain rates. Results establish the existence of a strain rate dependent ISE in the glassy polymer as hardness increases with decreasing the indentation depth. Employing the well-known shear transformation (ST) mediated flow theory and considering the statistical nature of the formation and distribution of the flow units, a rate dependent ISE model for glassy polymers is developed. This model is based on the possibility of the occurrence of discrete shear transformation sites within the deformed volume under the indenter which is controlled by the indentation depth and geometry. Results obtained from the proposed model show good agreement with the experimental values. The details considered in the model development in terms of the variations of indentation strain rate and elastic modulus with depth as well as validity and limitations of the model are discussed.

Rate-dependent size effects and material length scales in nanoindentation near the grain boundary for a bicrystal FCC metal

This paper addresses the size effects encountered in nanoindentation experiments of a copper bicrystal specimen in close proximity to the grain boundary. The experimental results show that as the distance r between the indentations and the grain boundary decreases, the hardness of the tested specimen increases. The single grain boundary in the bicrystal specimen is believed to play an important role to the increase in the hardness. The impact of the distance r on the hardness can be considered as a new type of size effect. The strain rate dependency of the length scale is investigated by incorporating a strain rate variable in the expression of the length scale. A temperature and rate-dependent indentation size effect model is used in order to determine the material intrinsic length scale using the corresponding data from the experiments. Nanoindentation experiments with different strain rates are performed in order to investigate the effect of the strain rate. In order to obtain more robust simulation results, the required numerical analysis is performed using ABAQUS Explicit software with a VUMAT user-subroutine which gives the physically based viscoplastic constitutive relations for the tested material.

Numerical investigation of nanoindentation size effect using micropolar theory

In this work, we investigate the properties of micropolar continua in the conical indentation of elastic–plastic material. The micropolar formulation is derived in axisymmetric condition, and a finite element model is implemented for the elastic–plastic contact solution of the indentation problem. It has been shown that micropolar description of the material is consistent with the Nix–Gao indentation size effect model at high indentation depth. The size effects obtained from micropolar continua are governed by hardening rules and yield function rather than the material elasticity. The numerical simulations have been verified by the experimental results coming from the open literature. Moreover, it has been shown that the proposed micropolar model allows the prediction of indentation size effect for both micro- and nanoindentations at the same time.

Depth-dependent stress–strain relation for friction prediction

The effect of strain gradient on the mechanical property of a material is implemented through a depth-dependent stress strain relation model in conventional finite element simulations for use in friction prediction. For the incorporation of a strain gradient effect, contact simulation involving asperities was developed with the assumption that the deformation pattern created by asperities from the tool surface in microforming is comparable to the deformation created by the indenter in a hardness test. Consequently, a depth-dependent stress–strain relation was derived from the indentation size effect model and this stress–strain relation was used in a simulation to show the effect of strain gradient on friction behaviour in microforming at different surface roughness levels. An experiment was conducted alongside the simulation and the results showed that with asperity ploughing considered as the major contributor to friction in microforming at room temperature, the simulation involving depth-dependent material properties is able to better predict the friction behaviour as compared to its continuum simulation counterpart.

Application of Indentation Size Effect in Finite Element Analysis for Friction Simulation

Contact simulation involving asperities was developed by assuming that the deformation by asperities is equivalent to the deformation by an indenter in a hardness test. Consequently, depth dependent flow stress curves were derived from the indentation size effect model from Abu Al-Rub and were used to simulate the influence of the number of asperities involved during contact on the distribution of contact pressure and the value of effective friction coefficient. Results from simulations suggested that multiplying the number of asperities in contact, when the size of the asperities is comparable to the size of the apparent contact, is not followed by proportional multiplication of the reaction forces. The competing phenomena observed in the simulation are then proposed as an explanation to friction size effect occurring in microforming.

Indentation size effect in bulk metallic glass

We systematically explored the indentation size effect (ISE), which is not expected to occur in non-crystalline materials due to the absence of dislocations and strain hardening, in bulk metallic glass (BMG). A series of nanoindentation experiments with different indenters result in somewhat surprising observations that show that ISE clearly does exist in BMG and can even be described by the ISE model for crystalline materials. The results are discussed in terms of possible mechanisms responsible for the ISE in BMG.

Nanohardness of molybdenum in the vicinity of grain boundaries and triple junctions

We performed a nanoindentation study of the inherent mechanical properties of grain boundaries and their triple junctions in molybdenum tricrystals of controlled geometry grown from the melt. A region with increased hardness is revealed in the vicinity of all grain boundaries studied. The width of this region is about 2 μm and the maximum relative increase of hardness at the grain boundary is about 30%. The local nanohardness at the triple junctions was found to be higher than that at the grain boundaries. This effect is analyzed in terms of the indentation size effect model of Nix and Gao, assuming that the grain boundaries represent ideal barriers for the first nucleated dislocation loops. We calibrated the parameters of the developed model using the data for nanohardness of grain boundaries. The model predicts values of nanohardness at the triple junctions which are in good agreement with the experimental results.

Indentation size effects in single crystal copper as revealed by synchrotron x-ray microdiffraction

For a Cu single crystal, we find that indentation hardness increases with decreasing indentation depth, a phenomenon widely observed before and called the indentation size effect (ISE). To understand the underlying mechanism, we measure the lattice rotations in indentations of different sizes using white beam x-ray microdiffraction (μXRD); the indentation-induced lattice rotations are directly measured by the streaking of x-ray Laue spots associated with the indentations. The magnitude of the lattice rotations is found to be independent of indentation size, which is consistent with the basic tenets of the ISE model. Using the μXRD data together with an ISE model, we can estimate the effective radius of the indentation plastic zone, and the estimate is consistent with the value predicted by a finite element analysis. Using these results, an estimate of the average dislocation densities within the plastic zones has been made; the findings are consistent with the ISE arising from a dependence of the dislocation density on the depth of indentation.

Indentation size effect in soft PZT ceramics with tetragonal structure close to the MPB

Nanoindentation measurements in tetragonal lead zirconate–titanate (PZT) ceramics doped with 0.8 mol% of La2O3 with a composition close to the morphotropic phase boundary (MPB) were performed using indenters with different tip radii. The elastic modulus was independent of indenter size but varied as expected with the Zr/Ti ratio of the samples. On the other hand, the calculated mean pressure remained constant for each indenter above the first pop-in, also varying with the Zr/Ti ratio, but was in all samples clearly dependent on the indenter radius. Indentation size effect (ISE) in tetragonal PZT was partially explained by the concept of ‘geometrically necessary dislocations’. However, the experimental data also indicate that the actual ISE model needs to be slightly extended to include the ferroelastic domain switching contribution.

Thickness Effects on the Plasticity of Gold Films

An indentation size effect is a common occurrence during nanoindentation. Thin and thick gold films, deposited using sputter deposition and evaporation, illustrate this at depths less than 100 nm. The indentation size effect, however, has been observed to be independent of film thickness. It has been modeled using a combination of an indentation size effect model and a parabolic hardening model. At the near surface regime, the indentation size effect model is dominant, and at larger depths, the parabolic hardening model is dominant, taking into effect the film thickness. The described model, which is a combination of these two, fits the experimental data for the sputter-deposited films and the evaporated films.

The correlation of the indentation size effect measured with indenters of various shapes

Experimental results are presented which show that the indentation size effect for pyramidal and spherical indenters can be correlated. For a pyramidal indenter, the hardness measured in crystalline materials usually increases with decreasing depth of penetration, which is known as the indentation size effect. Spherical indentation also shows an indentation size effect. However, for a spherical indenter, hardness is not affected by depth, but increases with decreasing sphere radius. The correlation for pyramidal and spherical indenter shapes is based on geometrically necessary dislocations and work-hardening. The Nix and Gao indentation size effect model (J. Mech. Phys. Solids 46 (1998) 411) for conical indenters is extended to indenters of various shapes and compared to the experimental results.