Indentation Yield Strength Research Articles

A comparative study on nanoscale mechanical properties of CrMnFeCoNi high-entropy alloys fabricated by casting and additive manufacturing

Additive manufacturing (AM) has emerged as a pioneering method for fabricating high entropy alloys (HEAs), yet a comprehensive comparison of their nanoscale mechanical properties with those produced by the conventional casting method remains unexplored. In this study, the nanoindentation was utilized to investigate the nanoscale elastic and plastic characteristics in both additive-manufactured (AM-ed) and as-casted single-phase face-centered cubic (FCC) equiatomic CrMnFeCoNi HEAs. Herein, the hardness, reduced modulus, indentation size effect (ISE), yield strength, fracture toughness, and strain rate sensitivity were comprehensively investigated. The results indicated that the hardness of AM-ed HEA was higher than the as-casted HEA, and the reduced modulus values showed no notable distinction between the two samples. The AM-ed HEA demonstrated simultaneous enhancements in yield strength and fracture toughness compared to the as-casted HEA. The as-casted HEA possessed a more distinct indentation size effect (ISE) than the AM-ed HEA. It was observed that the AM-ed HEA exhibited relatively lower strain rate sensitivity and a larger activation volume. This direct comparison of the mechanical properties and deformation mechanisms from a nanoscale view offers unique insights for optimizing and advancing AM techniques in the fabrication of HEAs.

Model for predicting temperature-dependent indentation yield strength and hardness considering size effects

The development of theoretical models to facilitate access to the mechanical properties of materials has been a long-standing pursuit of scholars. In this study, temperature-dependent yield strength models and temperature-dependent hardness models considering indentation size effects were developed under different loading modes (loading to the same displacement or the same load at different temperatures), respectively. Based on the Force-Heat Equivalence Energy Density Principle, the proposed models are free of any adjustable fitting parameters, and the quantitative relationships among temperature, yield strength/hardness, Young's modulus, and indentation load/displacement were successfully captured. All the available 36 groups of experimental data validate the accuracy of the proposed models over a wide temperature range. The yield strength and hardness of metallic materials over a wide temperature range can be easily predicted by using the developed models. Based on the parametric analysis of the models, it has been theoretically clarified for the first time that the two loading modes have opposite indentation size effects for the same material. Although the indentation size effect all decreases gradually with increasing temperature for different loading modes. The influence of loading modes should be noted in future studies where temperature-dependent indentation size effects need to be considered. This work systematically considers the effects of temperature, loading mode, indentation load/displacement, and Young's modulus on the indentation yield strength and hardness, which is important to facilitate the study of the deformation behavior of materials at different temperatures by indentation methods.

Semi-analytical model for predicting mechanical properties of metals using flat truncated cone indenter and its application

Based on the energy density equivalence method, a semi-analytical model describing the relationship between the material parameters of Hollomon’s law, load, displacement, and indenter geometry under flat truncated cone indentation (FTCI) was proposed. For a specific indenter, the model parameters were determined by presetting a small volume of material in finite element analysis (FEA). An FTCI test method to obtain the strain hardening coefficient and strain hardening exponent of the material from the load–displacement curve was proposed, and a damage factor (β) model was developed to predict the elastic modulus of the material by using multi-stage loading and unloading tests. Indentation tests were conducted on ten types of steel material using two types of flat truncated cone indenters with 15° and 30° angles. The results showed that the errors between the Young’s modulus measured via the indentation and uniaxial tensile tests were less than 5%; the relative errors between the indentation tensile strength and yield strength by the 30° flat truncated cone indenter and the uniaxial tensile test were less than 6%. For the 30° flat truncated cone indenter, the goodness-of-fit between the stress–strain relationship measured via the indentation and the uniaxial tensile tests was more than 0.95, and 80% of the goodness-of-fit was more than 0.97.

Microstructure and mechanical properties of 3D-printed dental Co-Cr alloys, produced by selective laser melting

Purpose The aim of present work was to investigate microstructure and mechanical properties of 3D printed by selective laser melting (SLM) Co- Cr alloys, intended for additive manufacturing in dentistry. Methods A scanning electron microscope (SEM), equipped with an integrated Energy-Dispersive X-Ray Spectroscopy (EDS) system was used for investigation of the surface morphology and elemental composition of the 3D- printed Co-Cr sample. The X-ray structural analysis of the 3D - printed Co-Cr sample was made with a Bruker D8 Advance powder X-ray diffractometer. An Atomic force microscopy (AFM) was used for investigation of the surface topography of the sample. Tensile test, a three-point bending test, and nanoindentation experiments were used for investigation of mechanical properties of the 3D-printed Co-Cr sample.The influence of two different strain rates (1 mm/min and 60 mm/min) on the flexural strength was investigated as well. Results Higher values of indentation hardness (6.76 GPa), tensile strength (1016 MPa), yield strength (636.5 MPa) and flexural strength (1908 and 1891 MPa) of the produced by selective laser melting Co-Cr alloys have been obtained compared to cast Co-Cr and Cr-Ni alloys. It was found that increasing the strain rate from 1 mm/min to 60 mm/min caused a proportional increase in recorded flexural strength of ~0.9%. Conclusions The obtained results showed that the laser-sintered Co-Cr alloy can fully replace the cast Co-Cr alloy in dentistry, regards as of its good mechanics properties, as well as the high precision of the final product.

In-situ synthesis of Ti5Si3-reinforced titanium matrix nanocomposite by selective laser melting: Quasi-continuous reinforcement network and enhanced mechanical performance

Titanium matrix nanocomposites (TMNCs) with quasi-continuously distributed Ti5Si3 reinforcements exhibit high material strength, good thermal stability, great tribological properties, and high fracture toughness. However, fabrication of such TMNCs via advanced additive manufacturing (AM) techniques has rarely been realized due to the presence of AM-induced large columnar grains and the cracking issue associated with the reinforcement coarsening. Here, we report a nanoparticle-mediated approach to in-situ fabricate nano-Ti5Si3 reinforced TMNC coatings by selective laser melting (SLM) of Ti powders and minor amount of SiC nanoparticles. Results showed that with the optimized SiC amount and SLM processing parameters, a crack-free and ultrahigh-strength TMNC consisted of near-equiaxed grain structure and nano-scale Ti5Si3 network at the grain boundaries was successfully produced. The optimized TMNC showed an ultrahigh surface microhardness of 706 VHN, 51.5% higher than that of SLM-fabricated SiC-free sample (466 VHN). Spherical nanoindentation results showed that the effective indentation modulus and indentation yield strength were improved by 62.6% and 57.2%, respectively. A more pronounced strain hardening phenomenon was also observed in the optimized TMNC. The dry sliding tests revealed that the wear rate was reduced by 70%, and the wear mechanism transferred from abrasion to adhesion.

Determination of peak ordering in the CrCoNi medium-entropy alloy via nanoindentation

Local chemical ordering (LCO) in the CrCoNi medium-entropy alloy was investigated by transmission electron microscopy (TEM) after different annealing treatments and their corresponding mechanical properties by bulk tensile tests and nanoindentation. A cold-rolled alloy was annealed at 1000°C for 0.5 h followed by ice water quenching and then aged at a number of different temperatures (600°C, 700°C, 800°C, 900°C, and 1000°C) under vacuum for 240 h to generate different degrees of chemical ordering. A splat-quenched sample rapidly cooled from the liquid phase was also examined. While bulk mechanical properties did not vary among samples with equivalent grain sizes, nanoindentation tests revealed notable differences. As indicated by the load at first pop-in using a Berkovich tip or the indentation yield strength via continuous stiffness measurements using a 10 μm spherical tip, the nanoindentation tests revealed that the stress for onset of plasticity during indentation varied with heat treatment and peaked in the 900°C aged sample. Energy-filtered TEM characterization indicated the presence of ordering in all specimens, with a higher degree of LCO in the aged samples relative to the splat-quenched and 1000°C-quenched samples. The evolution of LCO during aging was determined to occur on the time scale similar to those of bulk diffusion. The difference in nanoindentation strength was attributed to the difference in dislocation nucleation barriers imposed by different degrees of LCO.

Nanoindentation characteristics of nanocrystalline B2 CuZr shape memory alloy via large-scale atomistic simulation.

Nanoindentation tests are performed by molecular dynamics simulation to explore the mechanical properties of nanocrystalline B2 CuZr shape memory alloys with average grain sizes ranging from 6 to 18nm. Some paramount aspects are monitored, including indentation force-depth curve, hardness, yield strength, and elastic recovery. The results demonstrate an inverse Hall-Petch effect, i.e., the hardness decreases with the decrease in grain size. For the single crystalline B2 CuZr, dislocation nucleation and propagation are the major plastic mechanisms. However, grain cleavage, grain boundary compression, and grain rotation prevail over the plastic behaviors of nanocrystalline B2 CuZr alloys. The elastic recovery becomes stronger with the increase in grain size. Besides, the effects of temperature, indenter size, and indenter speed on the nanoindentation responses are evaluated quantitively.

Effect of C addition on microstructure and mechanical properties of laser micro-alloying Ti–Al–V–C titanium matrix composites

Most research activities involving the in-situ synthesis of TiC reinforced titanium matrix composites (TMCs) using laser additive manufacturing (LAM) focus on Ti and C interactions; however, interactions between other elements within the same context remain unexplored. This work details the design of a novel laser processing experiment called the laser micro-alloying (LMA) for depositing the Ti–Al–V-xC (0 ≤ x ≤ 1.0, wt.%) alloys, which was then used to elucidate the solid solution relationships between C, Al, and V in various matrices and the effect of C on the TiC in-situ precipitation behavior. The effect of C addition on the matrix microstructure was also elucidated to determine the alloying mechanism. It was confirmed that the solid solubility of C in the matrix decreases as Al and V contents increase. The TiC precipitation increases with increasing C content in the Ti alloy matrix, while the TiC morphologies in the samples evolved from feathery to clavate, then dendritic. It was noted that 0.12 wt.% C addition in Ti–4Al–3V resulted in a significant decrease in prior β grain size and α laths’ width, while further increasing C content had a negligible effect. Also, solid solution strengthening of C and continuous precipitation of TiC increased the indentation hardness, elastic modulus, and yield strength, with the increase being most significant when the C content is 0.12 wt.%. The work will lay a foundation for designing optimized TMCs used for LAM and other laser processing techniques.

Bayesian analysis of parametric uncertainties and model form probabilities for two different crystal plasticity models of lamellar grains in α+β Titanium alloys

The properties of individual phases and complex interactions of phases that affect mechanical behavior of metastable lamellar and Widmanstaetten morphologies of α+β Titanium alloys has long eluded quantitative assessment. As a result, the uncertainty of models for these complex multiphase morphologies is high, despite their importance in practical high-value applications. In this work, we present a comprehensive three-step framework for the estimation of intrinsic single crystal properties and the assessment of model plausibility for crystal plasticity models of lamellar α+β colonies of Titanium alloys using a dataset of experimentally obtained nanoindentation measurements on multiple grains in a polycrystalline sample. This approach does not rely upon submicron scale measurements within individual phases of defects, but rather exploits mesoscale measurements of collective response, making use of modern data science to assess uncertainties of both model form and model parameters. The first step of the three-step framework requires the establishment of a low-cost Gaussian Process Regression (GPR) surrogate (reduced order) model that predicts the indentation yield strength estimated by computationally expensive crystal plasticity finite element simulations. To reduce the number of finite element simulations necessary to establish the surrogate model, this work adopts an information-driven approach that selects additional simulations based on their potential to improve model accuracy. The second step involves the application of Markov Chain Monte Carlo (MCMC) sampling techniques in order to estimate the feasible intrinsic parameter configurations that are capable of capturing the trends in the experimental data. The sampling process provides rigorous uncertainty estimates for the intrinsic properties, in addition to providing an approximation for the model evidence. The model evidence is used to compute the relative probabilities of the chosen physics-based crystal plasticity model forms in the third step of the framework to enable model fusion. This framework is delineated and demonstrated in this paper through a case study involving two currently used crystal plasticity models for the colony morphology of Ti-6Al-4V.

Multiresolution investigations of thermally aged steels using spherical indentation stress-strain protocols and image analysis

In recent work, our research group has developed and demonstrated novel multi-resolution protocols capable of extracting indentation stress-strain (ISS) curves from tests on individual microscale constituents (e.g., phases, grains) as well as bulk properties of material microstructures. In addition, we recently developed protocols for design of consistent segmentation of micrographs. This work combines these recent advances in multi-resolution spherical indentation and image segmentation protocols to address the current challenges in the critical evaluation and advancement of physics-based composite models. These new research avenues are identified and demonstrated through a case study on thermally aged ferrite-pearlite steel samples, where the respective indentation yield strengths of the microscale constituents (i.e., ferrite, and pearlite) and the bulk yield strength of the samples were estimated from ISS tests measurements. The constituent volume fractions were extracted from segmented optical microscopy images. It is shown that the multi-resolution indentation yield strength and volume fraction measurements are highly consistent with the homogenization estimates from simple composite theories.

Characterization of distribution of residual stress in shot-peened layer of nickel-based single crystal superalloy DD6 by nanoindentation technique

Hardness, elastic modulus, yield strength, work hardening exponent, and residual stress within shot-peened layer of nickel-based single crystal superalloy DD6 were experimentally obtained by nanoindentation under a maximum load of 100 mN. The residual stresses calculated by different models have similar variation with the depth, but the values differ significantly with Wang's model most suitable for shot-peened DD6. It is found: the maximum residual compressive stress is 1.23 GPa at the depth of 40 μm; and the total depth of residual compressive stress layer is 140 μm. Mechanical properties such as indentation hardness and yield strength are affected by residual stress, which has little effect on elastic modulus. The effects of residual stress on various indentation parameters such as indentation work/displacement and contact depth/stiffness were quantified by linear expressions. The microstructural change in the shot-peened layer was found to be associated with the distribution of residual stress.

Efficacy of Graphene Nanosheets on the Plasma Sprayed Hydroxyapatite Coating: Improved Strength, Toughness and In-Vitro Bioperformance With Osteoblast

Graphene nanosheet (GNS) reinforced HA coating on Ti6Al4V substrate, fabricated using plasma spray, was developed for biomedical applications. Microstructural observation corroborated that the adding GNSs homogeneously distributed in the GNS/HA coating with typical lamellar structure featured by GNSs acting as binder between the discrete HA splats. Meanwhile, GNSs were observed to get in direct contact with HA matrix and form a clean GNS-HA interface without interfacial product. Results of various indentation tests showed simultaneous improvement in both indentation yield strength (~379 MPa) and fracture toughness (0.78±0.17 MPam1/2 ), mainly attributing to synergetic toughening and strengthening mechanisms associated with the adding GNSs such as load transfer, GNS pull-out, inter-layer sliding of a GNS, crack branching, GNS bridging and crack deflection. In addition, the GNS/HA coating exhibited improved biocompatible with MG-63 cell in terms of its attachment, adhesion strength, proliferation and differentiation. Hence, the GNS/HA composite coating with improved strength, toughness and enhanced biocompatibility makes it a promising candidate for bone regeneration and implantation.

Efficacy of graphene nanosheets on the plasma sprayed hydroxyapatite coating: Improved strength, toughness and in-vitro bioperformance with osteoblast

Graphene nanosheet (GNS) reinforced HA coating on Ti6Al4V substrate, fabricated using plasma spray, was developed for biomedical applications. Microstructural observation corroborated that the adding GNSs homogeneously distributed in the GNS/HA coating with typical lamellar structure featured by GNSs acting as binder between the discrete HA splats. Meanwhile, GNSs were observed to get in direct contact with HA matrix and form a clean GNS-HA interface without interfacial product. Results of various indentation tests showed simultaneous improvement in both indentation yield strength (~379 MPa) and fracture toughness (0.78 ± 0.17 MPam1/2), mainly attributing to synergetic toughening and strengthening mechanisms associated with the adding GNSs such as load transfer, GNS pull-out, inter-layer sliding of a GNS, crack branching, GNS bridging and crack deflection. In addition, the GNS/HA coating exhibited improved biocompatible with MG-63 cell in terms of its attachment, adhesion strength, proliferation and differentiation. Hence, the GNS/HA composite coating with improved strength, toughness and enhanced biocompatibility makes it a promising candidate for bone regeneration and implantation.

Critical Evaluation of Spherical Indentation Stress-Strain Protocols for the Estimation of the Yield Strengths of Steels

Accelerated development of advanced metals and alloys requires validated relationships between the high throughput indentation measurements and standardized tensile test protocols. The main objective of this paper is to critically evaluate the quantitative relationships between the yield strengths measured in standard tensile tests and the indentation yield strengths obtained from the recently developed spherical indentation stress-strain protocols for a broad range of steels with varying properties and microstructures. The tensile yield strengths for the tested steels varied from 300 to 1400 MPa, providing the broadest experimental evaluation to date for the spherical indentation stress-strain protocols. Standardized tension tests, standardized hardness measurements, and the recently developed spherical microindentation stress-strain protocols were employed on all the steels included in this study. The indentation stress-strain protocols provide a suitably normalized characterization of the material’s intrinsic mechanical response, when compared to standardized hardness measurements. It was found that the yield strengths estimated using the spherical indentation stress-strain protocols were within a 10% error. Furthermore, it was found that the tensile strengths estimated from the conventional hardness measurements exhibited higher error compared to those estimated from the spherical indentation stress-strain protocols. Spherical indentation stress-strain protocols were able to predict tensile yield strengths. The role of sample thickness on the indentation measurements was specifically investigated and quantified, resulting in specific recommendation for the minimum sample size for the indentation measurements.

Microstructural and Mechanical Characterization of Thin-Walled Tube Manufactured with Selective Laser Melting for Stent Application

This paper focuses on microstructural and mechanical characterization of metallic thin-walled tube produced with additive manufacturing (AM), as a promising alternative technique for the manufacturing of tubes as a feedstock for stents micromachining. Tubes, with a wall thickness of 500 μm, were made of 316L stainless steel using selective laser melting. Its surface roughness, constituting phases, underlying microstructures and chemical composition were analyzed. The dependence of hardness and elastic modulus on the crystallographic orientation were investigated using electron backscatter diffraction and nanoindentation. Spherical nanoindentation was performed to extract the indentation stress–strain curve from the load–displacement data. The obtained results were compared with those for a commercial 316L stainless steel stent. Both tube and commercial stent samples were fully austenitic, and the as-fabricated surface finish for the tube was much rougher than the stent. Microstructural characterization revealed that the tube had a columnar and coarse grain microstructure, compared to equiaxed grains in the commercial stent. Berkovich nanoindentation suggested an effect for the grain orientation on the hardness and Young’s modulus. The stress–strain curves and the indentation yield strength for the tube and stent were similar. The work is an important step toward AM of patient-specific stents.

Mechanical Responses of Primary-α Ti Grains in Polycrystalline Samples: Part I – Measurements of Spherical Indentation Stress-Strain Curves

Recently established spherical indentation stress-strain protocols have demonstrated the feasibility of measuring reliably the mechanical responses at different material structure length scales in a broad range of structural alloys. In the present study, we apply these high-throughput protocols on the primary α-phase grains in polycrystalline samples of Ti-5Al-2.5Sn, Ti-8Al-1Mo-1V, Ti-6Al-4V, Ti-6Al-2Sn-4Zr-2Mo and Ti-6Al-2Sn-4Zr-6Mo to aggregate a large experimental dataset that documents systematically the effects of α-phase chemical composition and grain orientation on the measured values of indentation modulus and the indentation yield strength. This dataset is being offered to the materials community in an open repository to allow further analyses of the effect of chemical composition of the α-phase on its single crystal elastic and plastic properties. This study establishes the feasibility and tremendous value of spherical indentation stress-strain protocols for documenting the grain-scale anisotropic mechanical responses of different α-phase compositions in high-throughput assays. In part II of this paper, the dataset is analyzed with advanced statistical approaches to estimate the single crystal elastic stiffness constants and the critical resolved shear strengths (CRSS) of the different Ti alloys studied.

On the Constraint Factor Relating Uniaxial and Indentation Yield Strength of Polycrystalline Materials Using Spherical Microindentation

Abstract High-throughput (HT) material property characterization has become increasingly important for the rapid and cost-effective development of new material systems. In recent years, spherical indentation using the Pathak-Kalidindi (P-K) protocol has been shown to be a reliable and HT method for capturing the elastic-plastic properties of several different types of materials. However, because of the higher hydrostatic pressure under the indenter tip and the constraint of the surrounding elastically deforming material, the indentation yield strength is higher than the uniaxial yield strength and they can be related by a scaling factor, referred to as the constraint factor. For fully plastic indentation, the constraint factor was found by Tabor to be ~2.8. However, the constraint factor is not well defined for elastic-plastic indentation and depends on the strain induced. In this work, the application of the P-K indentation protocol is demonstrated on a set of aluminum 7050 samples that have undergone different solution and aging treatments. Conventional uniaxial tensile and compression tests were also conducted on the same material. The constraint factor at the 0.2 % offset yield point was determined from the indentation and uniaxial measurements to be ~1.4. Experimental observations from our work and a similar work on aluminum 6061 showed a dependence of the constraint factor at the 0.2 % offset yield point on the uniaxial plastic properties. To explain these experimental observations, further investigation was carried out using finite element simulations of spherical indentation on materials with different yield strengths and hardening parameters. New insights are provided on the constraint factor at the 0.2 % yield point, which show that it increases with the ratio of the plastic to elastic strain. It was also found that the effect of the hardening is much less than that of the yield strength on the constraint factor at the 0.2 % yield point.

An investigation into mechanics and tribology of SnAgCu and MoO3 containing in 20CrMnTi based composites

Abstract The increasing in service life and operation reliability of the gearing system, the properties in mechanics and tribology of 20CrMnTi-steel-based materials need to be further optimized. Samples of 20CrMnTi (20CMT), 20CrMnTi-19.0 wt% SnAgCu (20CMT-S) and the 20CrMnTi-19.0 wt% SnAgCu-1.0 wt% MoO3 (20CMT-S-M) are fabricated on the D.R. Sinter® SPS3.20. Based on a nano-mechanical tribometer and the ball-on-disk tester, properties in mechanics and tribology of as-fabricated samples are understood. An excellent coordination of SnAgCu and MoO3 is beneficial in an increase in plasticity ductility from 9.5 to 10.5% of 20CMT-S-M. The values of indentation hardness, elasticity modulus and yield strength are slightly reduced down to 4.63 from 4.82 GPa, 168 from 200 GPa to 910 from 1030 MPa, respectively. An excellent lubrication of 20CMT-S-M is mainly attributed to a formed film containing SnAgCu and MoO3. An intra-lamellar separation of MoO3 tabular crystal can help the SnAgCu to better occur an outstanding plasticity deformation, that is more helpful in further decrease of friction resistance and material loss.

Nanoindentation Properties of 18CrNiMo7-6 Steel after Carburizing and Quenching Determined by Continuous Stiffness Measurement Method

In this work, the nanomechanical properties involving the indentation size effect (ISE) and yield strength of a surface-modified layer of 18CrNiMo7-6 steel after case hardening were investigated via nanoindentation experiments. The experimental results showed that the hardness increased with an increase in strain rate; the contact stiffness versus indentation depth curves take the form of upper convexity due to residual compressive stress relaxation. On the basis of the Ruiz-Moreno model, a modified model considering the cutoff parameter as a function of indentation depth was proposed. This model was able to better describe the ISE of the surface-modified layer. With the Hough transform error angle of 0.1° as the critical value (h0.1° is the corresponding depth), when h > h0.1°, the yield strength calculated by the Ma model started to disperse at the depth of h0.1°. These results provide useful insight into the local mechanical properties of 18CrNiMo7-6 steel after carburizing and quenching treatment.

Study of Local Mechanical Responses in an Epoxy–Carbon Fiber Laminate Composite Using Spherical Indentation Stress–Strain Protocols

Successful deployment of the highly heterogeneous, laminated, polymer matrix composites (PMCs) in high-performance structural applications is currently hindered by the lack of reliable experimental protocols for evaluation of the local mechanical responses at the salient meso-length/structure scales present in these material systems. Our main interest in this paper lies in establishing and demonstrating protocols for high-throughput evaluation of the local mechanical responses in PMCs at a length scale larger than the fiber diameter but smaller than the individual laminate (i.e., ply) thickness. This goal was accomplished in this work through a successful extension of the spherical indentation stress–strain protocols demonstrated recently for metallic samples. Specifically, plies with fibers at 0°, 30°, 60°, and 90° to the indentation direction were tested, and the means and standard deviations of their indentation moduli and the indentation yield strengths were measured and reported in this paper. The measured values of the indentation moduli were validated with finite element (FE) simulations performed using estimated values of the effective single laminate stiffness parameters. Furthermore, the measured variation in the indentation moduli was shown to correlate extremely well with the corresponding FE predictions that accounted for the measured variation in the local fiber volume fractions in the primary indentation deformed zones in the sample. These comparisons provided strong support for the validity of the extended spherical indentation protocols developed in this work for PMC samples.