Nix-Gao Model Research Articles

Characterization of Nanostructured and Ultrafine-Grain Aluminum-Silicon Claddings using the Nanoimpact Indentation Technique

The nanoimpact indentation technique is an emerging characterization technique that permits measurement of dynamic properties on a small scale. This article reports results on the characterization of nanostructured and ultrafine-grained Al-Si claddings using this technique. First, it was found that with this technique, the dynamic hardness of the material also becomes independent of the load, similar to nanoidentation, which yields to the concept of the existence of a dynamic true hardness. Second, the plasticity results have been compared to a strain-gradient plasticity model and have shown to deviate from the Nix-Gao model. Finally, a comparison between H/E and DH/E ratios has shown that the DH/E ratio correlates better with dry sliding wear results obtained for this material.

Nanoindentation hardness and its extrapolation to bulk-equivalent hardness of F82H steels after single- and dual-ion beam irradiation

The irradiation hardening behavior of reduced-activation ferritic steels after single Fe-ion beam irradiation and dual-ion (Fe ion and He ion) beam irradiation experiments was investigated with nanoindentation tests. The ion-irradiation experiments were conducted at 563K with 6.4MeV Fe3+ ions up to 3dpa at a 600nm depth from the irradiated surface. Furthermore, these experiments were conducted with and without simultaneous energy-degraded 1MeV He+ ions up to 300appm. The materials used were F82H, F82H+1Ni, and F82H+2Ni to investigate the effect of Ni addition on the irradiation hardening behavior. The measured nanoindentation hardness was converted to the bulk-equivalent hardness based on a combination of the Nix–Gao model to explain the indentation size effect and the composite hardness model to explain the softer substrate effect of the nonirradiated region beyond the irradiated depth range. It is clearly shown that the Ni addition enhances the irradiation hardening of F82H. The bulk-equivalent hardness is compared with the experimentally obtained Vickers hardness of F82H steels after neutron irradiation. The effect of simultaneously implanted helium on the irradiation hardening is negligible in the investigated irradiation conditions.

TEM analysis and determination of dislocation densities in nanostructured copper tube produced via parallel tubular channel angular pressing process

Parallel tubular channel angular pressing (PTCAP) is a recently developed novel intense plastic deformation method appropriate for fabrication ultrafine-grained and nanostructured cylindrical tubes. In the present work a commercially pure copper was processed via multi-pass PTCAP and the effects of number of passes on grain refinement and the dislocation density were studied. TEM analysis showed that after first pass elongated subgrains with interior tangled dislocations were formed. After pass number two the density of interior dislocations through the elongated grains was decreased. In the next stages of deformation, at pass number three, elongated grains almost disappeared and equiaxed grains with grain size about 150nm are formed as a result of dynamic recovery. The dislocation densities were measured by hardness indentation size effect using the Nix–Gao model. The results showed that increase in the number of PTCAP passes leads to decrease in the dislocation densities. The dislocation density is decreased to 2.48×109cm−2 after fourth passes from 18.1×109cm−2 after first pass. TEM results verified calculated values from the Nix–Gao model. Microhardness of the PTCAP processed tube through four passes increased to ∼142HV from initial value of about 62HV. Also, significant increase takes place after single pass and in the next passes the hardness value is saturated.

An Inverse Approach for Extracting Elastic–plastic Properties of Thin Films from Small Scale Sharp Indentation

An inverse method for extracting the elastic-plastic properties of metallic thin films from instrumented sharp indentation has been proposed in terms of dimensional analysis and finite element modeling. A wide range of materials with different elastic modulus, yield strength, and strain-hardening exponent were examined. Similar to the Nix-Gao model for the depth dependence of hardness H, ( H H 0 ) 2 = 1 + h H * h , the relationship between elastic modulus E and indentation depth h can be expressed as ( E E 0 ) 4 = 1 + h E * h . By combining these two formulas, we find that there is a relationship between yield stress σ y 0 and indentation depth h: σ y = σ y 0 ⋅ ( 1 + h H * h ) f ( n ) ⋅ ( 1 + h E * h ) [ 0.25 − 0.54 f ( n ) ] where σ y0 is the yield strength associated with the strain-hardening exponent n, the true hardness H0 and the true elastic modulus E0. f ( n ) = 1 2 ( 1 − n ) is constant, which is only related to n, and h H * and h E * are characteristic lengths for hardness and elastic modulus. The results obtained from inverse analysis show that the elastic-plastic properties of thin films can be uniquely extracted from the solution of this relationship when the indentation size effect has to be taken into account.

Size effects in the conical indentation of an elasto-plastic solid

The size effect in conical indentation of an elasto-plastic solid is predicted via the Fleck and Willis formulation of strain gradient plasticity (Fleck, N.A. and Willis, J.R., 2009, A mathematical basis for strain gradient plasticity theory. Part II: tensorial plastic multiplier, J. Mech. Phys. Solids, 57, 1045–1057). The rate-dependent formulation is implemented numerically and the full-field indentation problem is analyzed via finite element calculations, for both ideally plastic behavior and dissipative hardening. The isotropic strain-gradient theory involves three material length scales, and the relative significance of these length scales upon the degree of size effect is assessed. Indentation maps are generated to summarize the sensitivity of indentation hardness to indent size, indenter geometry and material properties (such as yield strain and strain hardening index). The finite element model is also used to evaluate the pertinence of the Johnson cavity expansion model and of the Nix–Gao model, which have been extensively used to predict size effects in indentation hardness.

Analysis of microbending of CuZn37 brass foils based on strain gradient hardening models

Microbending experiments of brass foils were conducted and demonstrated clear size effects, i.e., the normalized bending moment increased with the reduction of foil thickness. The experiments were modeled using the classical plasticity model and three strain-gradient plasticity models. A modified Nix–Gao model was proposed to consider the number of twins across the thickness direction in strain hardening. It was found that the proposed model could better predict the normalized bending moment for both fine-grain and coarse-grain foils and the proposed equation for material intrinsic length could better capture the physics of material deformation. Furthermore, micro-hardness distributions in the bending area were measured and obvious lower-hardness region was found in the middle layer of fine-grain foils instead of coarse-grain foils. This indicates that the modified Nix–Gao model and the assumption of fully plastic bending worked better for coarse-grain foils than for fine-grain foils.

Mechanical behavior of Ti cold spray coatings determined by a multi-scale indentation method

Abstract The effect of the indentation load on the hardness measurements on cold spray Ti coatings was evaluated. Varied load indentation was carried out with nanoindentation at loads between 1 and 20 mN and microindentation at loads between 0.1 and 5 N. Nanoindentation measurements showed an indentation size effect and were fit to the Nix–Gao model for strain gradient plasticity. The microindentation measurements depended on the indentation size in a manner different from the Nix–Gao model and were strongly affected by the material porosity and cohesive strength between cold sprayed particles. By comparing results from the two indentation techniques, a hardness loss parameter was formulated that helped explain the mechanical behavior at all length scales and its relation to the various defects within the coating. Comparisons of the hardness loss parameter between cold spray coatings and a bulk Ti plate showed that this new parameter provided a good measure of when the mechanical behavior of the cold spray coatings approached that of a traditionally manufactured material. Evaluation of the hardness loss parameter of coatings and the bulk material, combined with microstructural characterization before and after indentation, was used to examine the relationships between processing conditions, coating microstructure and mechanical behavior for coatings fabricated with spherical and non-spherical Ti powders.

Dislocation structures in nanoindented ductile metals—a transmission electron microscopy direct observation

Several attempts have been made over recent years to understand indentation size effect (ISE). The theoretical models, based essentially on strain gradient plasticity theories, such as the Nix–Gao model, predict that ISE is caused by an increase in the density of dislocations as the indentation size decreases. Molecular dynamics simulation results tend to confirm this fact, but the truth is that very few experimental studies exist on the direct observation of how dislocations are generated and accommodated in the vicinity of nanoindentations. In this study, using a Ni transmission electron microscopy thin foil as model material, we show that when the material is submitted to atomic force microscopy-based nanoindentation a high dislocation density zone is generated at the centre of the indented region, and that prismatic loop and helical dislocations are emitted sidewards from the central region of the nanoindentation along the ⟨1 1 0⟩ directions. Moreover, we show that the dislocation array formed during the nanoindentation process is far from the ideal model proposed by Nix and Gao, based on load axi-centred dislocation loops. With this study we aim at contributing to a better comprehension of ISE mechanisms in ductile metals.

A new approach to evaluate irradiation hardening of ion-irradiated ferritic alloys by nano-indentation techniques

The present work investigates the irradiation hardening of Fe-based model ferritic alloys after Fe-ion irradiation experiments in order to deduce mechanistically based nominal hardness from the nano-indentation tests on the ion-irradiated surface. Ion-irradiation experiments were carried out at 290 °C with 6.4 MeV Fe 3+ ions. The constant stiffness measurement (CSM) was used to obtain the depth-profile of hardness. The results has been analyzed and discussed based on the Nix–Gao model and an extended film/substrate system hardness model. The depth-sensing nano-indentation techniques with CSM revealed that the hardness gradient of the unirradiated Fe-based model alloy can be explained through the indentation size effect (ISE). On the other hand, the gradient of ion-irradiated surface of these samples includes not only the ISE but also softer substrate effect (SSE). We propose a new approach to evaluate a nominal hardness, which may connect to the bulk hardness, from experimentally obtained nano-hardness depth profile data.

Mechanical properties of Y 2Ti 2O 7

The mechanical properties of Y2Ti2O7 ceramics have been systematically characterized. Y2Ti2O7 shows higher hardness (12.1 GPa) than Y2O3 and TiO2. The Young’s modulus of Y2Ti2O7 (262 GPa) is close to that of TiO2 and 50% higher than that of Y2O3. The fracture toughness (1.9 MPa m1/2) and bending strength (206 MPa) are comparable with those of Y2O3 but half those of TiO2. The indentation size effect on the hardness in nanoindentation tests is discussed based on the Nix–Gao model.

Dislocation-based model for predicting size-scale effects on the micro and nano indentation hardness of metallic materials

In micro- and nano-indentation tests for evaluating strength and stiffness properties of engineering materials, a commonly observed phenomenon is the dependence of material properties on the indent size, also known as indentation size effect (ISE). The objective of the present work is to formulate a micro-mechanical based model based on dislocation mechanics for predicting ISE from conical or pyramidal (Berkovich and Vickers) indenters and to compare it with the most widely used Nix-Gao model. The key idea proposed here while deriving the model is a non-linear coupling between the geometrically necessary dislocations (GNDs) and the statistically stored dislocations (SSDs) that eventually allows it to simultaneously predict ISE from both micro- and nano-indentations tests on a wide range of metallic materials while the Nix-Gao model fails to do so. The work also presents a method for identifying the length scale parameter from micro- and nano-indentation experiments and also correlates it with the spacing between dislocations and thus gives a physical interpretation of the material intrinsic length scale.

Determination of dislocation density from hardness measurements in metals

For the first time the Nix–Gao model for indentation size effect (ISE) is used to estimate the dislocation density in a metal. The estimate of dislocation density obtained by this method, using Ni as a case study, is compared with the values obtained from direct observation by transmission electron microscopy. It is shown that the estimate of dislocation density from indentation hardness measurements, adjusted by the Nix–Gao model, gives values consistent with those obtained by TEM, provided that the proper procedures to minimize errors are adopted. Although the direct observation of dislocations by TEM gives additional structural information, the indirect method to estimate dislocation density based on hardness measurements is more efficient, since the sample preparation method, measurement procedure and analysis of results are easier and faster.

Microstructural investigation of the volume beneath nanoindentations in copper

The deformed volume below nanoindentations in copper single crystals with a 〈 1 1 ¯ 0 〉 { 1 1 1 } orientation is investigated. Using a focused ion beam workstation, cross-sections through nanoindentations were fabricated and examined using the electron backscatter diffraction technique. Additionally a transmission electron microscopic foil through the middle of an imprint was prepared and analysed. Due to changes in the crystal orientation around and beneath the indentations the plastically deformed zone can be visualized and compared with the measured hardness values. Furthermore, the hardness data were analysed in terms of geometrically necessary dislocations using the Nix–Gao model, where a linear relationship was found for H 2 vs. 1/ h c, but with different slopes for large and shallow indentations. The orientation “micrographs” indicate that this behaviour is associated with a change in the deformation mechanism. Consequently, two models based on possible dislocation arrangements are presented and compared with the experimental findings. For large indentations a dislocation pile-up model similar to those used to explain the Hall–Petch effect is suggested, while the model for shallow imprints uses far-reaching dislocation loops to accommodate the shape change of the indenter.

Prediction of micro and nanoindentation size effect from conical or pyramidal indentation

It is well-known by now that the hardness of the material at the micron and submicron length scales is dependent on the indent size. The objective of this work is to formulate a micromechanical-based model that can be used to predict simultaneously the indentation size effect (ISE) from both micro and nanoindentations by conical or pyramidal (Berkovich and Vickers) indenters. This model is based on the evolution of geometrically necessary dislocations (GNDs) beneath the indenter which is nonlinearly coupled to the evolution of statistically stored dislocations (SSDs) through the Taylor’s hardening law. It is shown through comparisons with micro and nanoindentation experimental data that the proposed model gives much better predictions of hardness at small indentation depths as compared to the Nix–Gao model. It is concluded that when using the Taylor’s hardening law a simple sum of flow stresses from SSDs and GNDs is more adequate than the simple sum of SSD and GND densities. Moreover, it is shown that the length scale responsible for the ISE is proportional to the spacing between dislocations. Thus, it is concluded that materials with smaller length scales are harder but exhibit lower ISE.

Microstructural Investigation of the Deformation Zone below Nano-Indents in Copper

AbstractThe deformation zone below nanoindents in copper single crystals with an <110>{111} orientation is investigated. Using a focused ion beam (FIB) system, cross-sections through the center of the indents were fabricated and subsequently analyzed by means of electron backscatter diffraction (EBSD) technique. Additionally, cross-sectional TEM foils were prepared and examined. Due to changes in the crystal orientation around and beneath the indentations, the plastically deformed zone can be visualized and related to the measured hardness values. Furthermore, the hardness data were analyzed using the Nix-Gao model where a linear relationship was found for H2 vs. 1/hc, but with different slopes for large and shallow indentations. The measured orientation maps indicate that this behavior is presumably caused by a change in the deformation mechanism. On the basis of possible dislocation arrangements, two models are suggested and compared to the experimental findings. The model presented for large imprints is based on dislocation pile-ups similar to the Hall-Petch effect, while the model for shallow indentations uses far-reaching dislocation loops to accommodate the shape change of the imprint.

Indentation study of titanium, zirconium, and hafnium hydrides

AbstractMechanical properties of titanium, zirconium, and hafnium hydrides were evaluated by indentation tests in the present study. Single-phase bulk specimens of titanium, zirconium, and hafnium hydrides with a fluorite type structure were produced using a Sieverts' apparatus. The indentation hardness values of titanium and hafnium hydrides were lower than those of the pure metals and drastically decreased with increasing hydrogen content. The hardness of zirconium hydride was higher than that of pure zirconium and slightly depended on hydrogen content. The hardness of the hydrides decreased with increasing maximum indentation load, which has been referred to as indentation size effect. The dependences of hardness on the indentation depth for the hydrides were found to obey the Nix–Gao model. The indentation size effects for the hydrides were significantly smaller than those for the pure metals.

Influence of tip bluntness on the size-dependent nanoindentation hardness

Abstract Nanoindenter tip bluntness modifies the size-dependent hardness relationship given by the Nix–Gao model. At shallow depths, fewer geometrically necessary dislocations are needed to accommodate the blunted tip. The experimentally measured ratio of plastic to total contact depth decreases and the hardness decreases relative to the idealized Nix–Gao model.