Theory of indentation of piezoelectric materials

A general theory is presented for the axisymmetric indentation of piezoelectric solids within the context of fully coupled, transversely isotropic elasticity models. Explicit expressions for P–h curves are derived for spherical, conical as well as cylindrical punch indenter geometries in a manner that can be directly related to the experimental measurements. In addition, results for different electrical boundary conditions that employ conducting or insulating indenters are also presented. The theory reveals that the indentation load vs penetration depth, and the contact area vs penetration depth relations have the same mathematical structure as the classical elastic indentation problem. It is, however, demonstrated that the electric field induced during indentation as a result of the electrical–mechanical coupling can resist or aid in the penetration of the indenter into the piezoelectric material depending on the electrical conductivity of the indenter and the surface boundary conditions of the indented substrate. It is also shown that the piezoelectric material exhibits pile-up or sink-in of material around the indenter as a consequence of electromechanical coupling, despite the absence of any inelastic deformation processes or strain hardening. The theoretical predictions are corroborated with detailed finite-element simulations for different indenter geometries. The theoretical results facilitate the prediction of some transient electrical effects which can be used in conjunction with experiments for the estimation of some of the elastic, dielectric and piezolectric constants during instrumented indentation. Specific examples and details of such applications are addressed in separate papers.     

压痕诱发单晶Si非晶化相变区的三维空间分布结构

利用航向电子显微术（ＴＥＭ）观察了单晶Ｓｉ维氏压痕区的结构转变。平视和截面像相结合给出了压痕诱导Ｓｉ非昌化的轮廓图。其区域为与压头相似的倒四棱锥金字塔形。但其棱边夹角小于相对应的压头相对棱夹角，残留的准确无误痕深度很浅。直接证明了Ｓｉ的压痕区具有很大的弹性回复量     

Combining indentation and diffusion couple techniques for combinatorial discovery of high temperature shape memory alloys

We demonstrate the possibility of accelerated identification of potential compositions for high-temperature shape memory alloys (SMAs) through a combinatorial material synthesis and analysis approach, wherein we employ the combination of diffusion couple and indentation techniques. The former was utilized to generate smooth and compositionally graded inter-diffusion zones (IDZs) in the Ni–Ti–Pd ternary alloy system of varying IDZ thickness, depending on the annealing time at high temperature. The IDZs thus produced were then impressed with an indenter with a spherical tip so as to inscribe a predetermined indentation strain. Subsequent annealing of the indented samples at various elevated temperatures, T_a, ranging between 150 and 550 °C allows for partial to full relaxation of the strain imposed due to the shape memory effect. If T_a is above the austenite finish temperature, A_f, the relaxation will be complete. By measuring the depth recovery, which serves as a proxy for the shape recovery characteristic of the SMA, a three-dimensional map in the recovery–temperature–composition space is constructed. A comparison of the published A_f data for different compositions with the T_a data shows good agreement when the depth recovery is between 70% and 80%, indicating that the methodology proposed in this paper can be utilized for the identification of promising compositions. Advantages and further possibilities of this methodology are discussed.     

Size dependence of radiation-induced amorphization and recrystallization of synthetic nanostructured CePO4 monazite

Monazite, CePO₄, is considered an important phosphate-type structure for the incorporation and disposal of actinides. Nanocrystalline monazite with particle sizes ranging from 20 nm to greater than 100 nm was synthesized. The displacive and ionizing effects of radiation were investigated, separately and simultaneously, for 1 MeV Kr²⁺ and 200 keV electron irradiations. In situ transmission electron microscopy observations indicated that CePO₄ nanoparticles can be readily amorphized by 1 MeV Kr²⁺ irradiation. Nanostructured monazite displays greater critical amorphization doses and lower critical temperatures than that of bulk natural monazite, suggesting enhanced amorphization tolerance. A strong size dependence on radiation-induced amorphization was observed in which the smaller-sized particles (20 nm) are less resistant to amorphization compared with larger-sized particles (40 nm). The excess surface energy of nanostructured materials, as suggested by the larger surface area upon the reduction of particle size, may alter the energy difference between amorphous and crystalline phases, thus affecting the radiation stability. With 200 keV electron-beam irradiation, CePO₄ previously amorphized by 1 MeV Kr²⁺ experienced an ionizing-radiation-induced recrystallization. A greater recrystallization rate was observed for the smaller-sized particles. Under simultaneous electron and displacive ion irradiations, CePO₄ displayed greater tolerance against amorphization, probably as a result of radiation-induced recovery of displacive damage by the ionizing radiation. The strong size dependence of displacive radiation-induced amorphization and ionizing-radiation-enhanced recrystallization processes for nanostructured CePO₄ implies that an optimized size regime may exist in which nanostructured materials are more tolerant of radiation-induced amorphization.     

Cross-sectional electron microscopy observation on the amorphized indentation region in [001] single-crystal silicon

The amorphized region of single-crystal silicon (c-Si) induced by Vickers indentation has been studied cross-sectionally by transmission electron microscopy (TEM) and high-resolution electron microscopy (HREM). A comparison between the V-shaped profile of the amorphous region and the stress isobars under the indenter shows that the deviatoric stress plays a significant role in the formation of amorphous silicon (a-Si). A number of defects near the crystalline/amorphous (c/a) interface, and the refinement and rotation of grains at local regions, are observed by HREM. The distortion of lattice fringes in the c-Si region and the domains characterized by distorted lattice in the a-Si region near the interface as well as continuous transition from the crystalline to the amorphous region at the interface are also observed. A possible mechanism of defect-induced or heavy-deformation-induced amorphization of silicon under indentation is suggested.     

Nanoindentation of porous bulk and thin films of La0.6Sr0.4Co0.2Fe0.8O3−δ

In this paper we show how reliable measurements on porous ceramic films can be made by appropriate nanoindentation experiments and analysis. Room-temperature mechanical properties of the mixed-conducting perovskite material La_0.6Sr_0.4Co_0.2Fe_0.8O_3?δ (LSCF6428) were investigated by nanoindentation of porous bulk samples and porous films sintered at temperatures from 900 to 1200 °C. A spherical indenter was used so that the contact area was much greater than the scale of the porous microstructure. The elastic modulus of the bulk samples was found to increase from 33.8 to 174.3 GPa and hardness from 0.64 to 5.32 GPa as the porosity decreased from 45% to 5% after sintering at 900–1200 °C. Densification under the indenter was found to have little influence on the measured elastic modulus. The residual porosity in the “dense” sample was found to account for the discrepancy between the elastic moduli measured by indentation and by impulse excitation. Crack-free LSCF6428 films of acceptable surface roughness for indentation were also prepared by sintering at 900–1200 °C. Reliable measurements of the true properties of the films were obtained by data extrapolation provided that the ratio of indentation depth to film thickness was in the range 0.1–0.2. The elastic moduli of the films and bulk materials were approximately equal for a given porosity. The 3-D microstructures of films before and after indentation were characterized using focused ion beam/scanning electron microscopy tomography. Finite-element modelling of the elastic deformation of the actual microstructures showed excellent agreement with the nanoindentation results.     

Finite element analysis for conical indentation unloading of elastoplastic materials with strain hardening

Finite element analysis is conducted for various elastoplastic solids with linear strain-hardening that are contacted by cone indenters having three different inclined face angles [β=10.0°, 19.7° (Vickers equivalent angle), and 30.0°]. The indentation load P vs. penetration depth h relationship during unloading is intensively examined. Although the P–h unloading relationship for a cone indentation is apparently non-quadratic [P=α(h−h_r)^m; m<2.0], it is emphasized from the geometrical similarity of cone indentation that the unloading process conforms to a quadratic relation [P=k₂(h−h_r)²; h_r is the residual depth of impression after a complete unload] in its essential physical process. The unloading parameter k₂ is, then, directly related to the elastic modulus E^′ of the material on the basis of the concept of “effective face angle β_eff” of indenter. The indentation unloading process is well described by the loading process of a conical indenter with the effective face angle of β_eff(=β−β_r) pressed into a flat elastic half-space, in which the inclined face angles of the indenter used and of the residual hardness impression formed are defined by β and β_r, respectively. The non-quadratic term included in unloading P–h relation results from the locally distorted convex surface profile of the residual impression.     

A continuum dislocation model of Vickers indentation on a zirconia

The Vickers indentation of 3 mol% Y₂O₅ partially stabilized ZrO₂ (3Y-PSZ) was examined by a depth sensing technique and analyzed on a continuum dislocation model. The model is based on the punching of prismatic dislocation loops to accommodate the volume of plastic penetration at the prismatic indentation. A procedure is proposed for solving an inverse problem to estimate the plastic core zone configuration and yield stress in the indentation of brittle materials using the information obtained from the experimental indentation curve of the material. It was found that there are an infinite number of solutions as a function of the plastic zone configuration. The most appropriate solution was obtained by comparing the predicted profiles of the indentation with the profile observed by a topographic scanning electron microscope. The estimated plastic zone configuration and yield stress show reasonable agreement with experimental data of Y-PSZ in the literature.     

Formation of nanocrystalline structure in the amorphous Ti50Ni25Cu25 alloy upon severe thermomechanical treatment and the size effect of the thermoelastic martensitic B2 ? B19 transformation

The results of the comparative analysis of the Ti₅₀Ni₂₅Cu₂₅-alloy structures produced in the initial amorphous state by rapid quenching from the melt (RQM), after severe plastic deformation by torsion under high pressure (HPT), and postdeformation heat treatment (PHT) are presented. The study was carried out using neutron and X-ray diffraction, transmission and scanning electron microscopy, and measurements of electrical properties. The initially amorphous alloy has been established to nanocrystallize after torsion under a pressure of 7 GPa to 0.5 revolutions of the anvil. Then, after 1, 5, 10, and 15 rev, the alloy again undergoes the strain-induced amorphization even with the retention, even after 5–15 rev, of a large number of highly dispersed nanocrystals less than 3–4 nm in size with a distorted B2 lattice in the amorphous matrix. Their crucial role as nuclei of crystallization provides the total low-temperature nanocrystallization during subsequent annealing starting from 250–300°C. It is shown that PHT of the alloy amorphized by HPT makes it possible to produce extremely uniform nanocrystalline (NC), submicrocrystalline (SMC), or bimodal (NC + SMC) austenitic B2-type structures in it. A complete diagram of thermoelastic martensitic transformations in the region of B2-austenite states, from nanostructured state to conventional polycrystalline one, has been constructed. The size effect on the stabilization of martensitic transformation in nanocrystalline B2 alloy has been established.     

Deformation in a Zr57Ti5Cu20Ni8Al10 bulk metallic glass during nanoindentation

Nanoindentation experiments of a Zr₅₇Ti₅Cu₂₀Ni₈Al₁₀ bulk metallic glass were performed with indentation loads ranging from 200 to 2000 μN. Both the indentation hardness and the reduced contact modulus decreased with the increase in the indentation load due to the propagation of shear bands underneath the indenter – the occurrence of a softening effect. The ratio of the indentation hardness to the reduced contact stiffness was a function of the reciprocal of the indentation depth. Based on the concept of diffusion-induced stresses, a one-dimensional constitutive relation between the change of the excessive free volume and the flow stress was proposed. The indentation-size effect as observed in the indentation tests was explained through the consideration of the contribution of the strain gradient in the constitutive relation.     

Formation of the nanocrystalline structure in the Ti<Subscript>50</Subscript>Ni<Subscript>25</Subscript>Cu<Subscript>25</Subscript> shape-memory alloy under severe thermomechanical treatment

Results of comparative studies of the structure of the cast martensitic Ti₅₀Ni₂₅Cu₂₅ alloy in the initial state, after severe plastic deformation by high-pressure torsion (HPT), and after subsequent annealing are presented. The studies have been performed by X-ray diffraction, transmission and scanning electron microscopy, and measurements of electrical properties. It has been established that the alloy undergoes almost complete amorphization after torsion using 5 and 10 rev of anvils under a pressure of 7 GPa. This result can be explained by the large value of shear deformation (true strain from 6 to 7 units) and the retention of an extremely large quantity of highly dispersed (less than 3–4 nm in size) nanocrystals with a distorted B2 lattice in the amorphous matrix even at room temperature. Their determining role as nuclei of crystallization ensures the total process of low-temperature nanocrystallization upon subsequent annealing, beginning from 250–300°C. It is shown that the annealing of the alloy amorphized during HPT makes it possible to produce extremely uniform nanocrystalline (NC), submicrocrystalline (SMC), or bimodal (NC + SMC) structures of B2 austenite. For the first time, a complete diagram of thermoelastic martensitic transformations in the field of B2-austenite states, from nanostructured to usual polycrystalline, has been constructed for the Ti₅₀Ni₂₅Cu₂₅ alloy. The size effect of stabilization of the martensite transformation has been found in the nanocrystalline B2 alloy.     

Depth-sensing instrumented indentation with dual sharp indenters

A methodology for interpreting instrumented sharp indentation with dual sharp indenters with different tip apex angles is presented by recourse to computational modeling within the context of finite element analysis. The forward problem predicts an indentation response from a given set of elasto-plastic properties, whereas the reverse analysis seeks to extract elasto-plastic properties from depth-sensing indentation response by developing algorithms derived from computational simulations. The present study also focuses on the uniqueness of the reverse algorithm and its sensitivity to variations in the measured indentation data in comparison with the single indentation analysis on Vickers/Berkovich tip (Dao et al. Acta Mater 49 (2001) 3899). Finite element computations were carried out for 76 different combinations of elasto-plastic properties representing common engineering metals for each tip geometry. Young’s modulus, E, was varied from 10 to 210 GPa; yield strength, σ_y, from 30 to 3000 MPa; and strain hardening exponent, n, from 0 to 0.5; while the Poisson’s ratio, ν, was fixed at 0.3. Using dimensional analysis, additional closed-form dimensionless functions were constructed to relate indentation response to elasto-plastic properties for different indenter tip geometries (i.e., 50°, 60° and 80° cones). The representative plastic strain ε_r, as defined in Dao et al. (Acta Mater 49 (2001) 3899), was constructed as a function of tip geometry in the range of 50° and 80°. Incorporating the results from 60° tip to the single indenter algorithms, the improved forward and reverse algorithms for dual indentation can be established. This dual indenter reverse algorithm provides a unique solution of the reduced Young’s modulus E¹, the hardness p_ave and two representative stresses (measured at two corresponding representative strains), which establish the basis for constructing power-law plastic material response. Comprehensive sensitivity analyses showed much improvement of the dual indenter algorithms over the single indenter results. Experimental verifications of these dual indenter algorithms were carried out using a 60° half-angle cone tip (or a 60° cone equivalent 3-sided pyramid tip) and a standard Berkovich indenter tip for two materials: 6061-T6511 and 7075-T651 aluminum alloys. Possible extensions of the present results to studies involving multiple indenters are also suggested.     

Gradients in elastic modulus for improved contact-damage resistance. Part I: The silicon nitride–oxynitride glass system

Silicon nitride (Si₃N₄)-based graded materials were fabricated with controlled, unidirectional gradients in elastic modulus from the surface to the interior. This was accomplished by infiltrating a low modulus silicon oxynitride glass into a dense, higher modulus, Si₃N₄ ceramic. Elastic Hertzian indentation (spherical indenter) experiments were performed on both the graded and the monolithic Si₃N₄. While Hertzian indentation of the monolithic ceramic resulted in classical cone cracks, such cracks were completely suppressed in the graded materials at comparable load levels, despite the lower strength and lower toughness of the surface layer comprising glass. Finite element analysis (FEA) of the stresses associated with the indentation was also performed to gain insight into the mechanism for the enhanced contact damage resistance in the graded materials. The computational analysis revealed that the maximum tensile stresses outside the Hertzian contact circle, which drive the cone-cracks, are reduced by approximately 30% relative to those present in the monolithic silicon nitride. This reduction in the tensile stresses more than compensates for the lower toughness at the graded material surfaces, relative to the monolithic Si₃N₄. The FEA also allowed us to develop some strategies for elastic–modulus-gradients that would lead to further improvements in the cone-crack suppression characteristics of graded materials in general.     

Indentation modulus and hardness of polyaniline thin films by atomic force microscopy

Polyaniline (PANI) thin films with different thicknesses have been deposited on indium tin oxide (ITO) coated glass substrates by electrochemical polymerization of the aniline monomer in H₂SO₄ aqueous solution. By using the tip of an atomic force microscopy (AFM) apparatus as an indenter, cantilever deflection versus sample vertical displacement curves have been acquired and analyzed for evaluating the contact stiffness, by using an approach analogous to that developed for standard depth sensing indentation (DSI) tests. After the calibration performed using a set of polymeric reference materials, indentation modulus and hardness of PANI films have been deduced as a function of the reached maximum penetration depth. By using a model originally proposed for the analysis of standard DSI measurements, indentation modulus and hardness values of only PANI are finally deduced from the corresponding apparent values measured for the film-substrate systems, although they have to be considered as semi-quantitative estimations, since the roughness of the films does not allow a certain determination of the local thickness in correspondence of the probed points.     

Processing,microstructure and mechanical properties of HfB2-ZrB2-SiC composites: Effect of B4C and carbon nanotube reinforcements

The fully dense HfB₂-ZrB₂-SiC composites were processed using spark plasma sintering (SPS) at 1850 °C. The effect of reinforcements (B₄C and CNT) on the densification as well as mechanical properties were investigated and compared (with monolithic) in the present study. The study showed that the addition of B₄C and CNT were not only beneficial for the densification but also towards enhancing the mechanical properties (hardness, elastic modulus, and fracture toughness) of HfB₂-ZrB₂-SiC composites. The augmentation in the mechanical properties establish the synergy between solid solution formation (with the equimolar composition of HfB₂/ZrB₂) and the reinforcements (SiC, B₄C, and CNT). The highest increase in the indentation fracture toughness with the reinforcements of B₄C as well as CNT is >3 times (~13.8 MPam^0.5 when it is 3–4 MPam^0.5 for monolithic ZrB₂/HfB₂) on HfB₂-ZrB₂-SiC composites, which is attributed to the crack deflection and pull-out mechanisms. An increase in the analytically quantified interfacial compressive residual stresses in the composites during SPS processing with the synergistic addition of reinforcements (SiC, B₄C, and CNT) and its effect on the indentation fracture toughness has also been addressed.     

Simulation and Experiment on Surface Morphology and Mechanical Properties Response in Nano-Indentation of 6H-SiC

The nano-indentation test for 6H-SiC is carried out with a Berkovich indenter. The indentation surface morphology is analyzed by SEM, which show that when the maximum load P _max is 8 mN, there is only plastic deformation and no cracks on the surface of workpiece after unloading process, and when P _max is 10 mN, there is the initiation of crack occurring on the surface of workpiece after unloading process. Based on the strain hardening model, the three-dimensional finite element method of nano-indentation for 6H-SiC is carried out. Simulation results show that in the unloading process the maximum stress and the maximum strain occur in the contact area between the workpiece with the indenter edges, which is consistent with the experimental results. When propagate to the surface from the subsurface, the cracks are subjected to the type I stress and the type II stress due to elastic recovery. After propagating to surface of workpiece, the cracks propagate along a fixed direction because the proportion of type I stress is much larger than that of type II stress. The influence of the cleavage plane on the propagation direction of cracks is obvious. The cracks propagate more easily when the indenter edges are along cleavage plane. The indentation depth and residual depth increase with the increase of P _max. While, the elastic recovery rate gradually decreases and tends to be stable with the increase of P _max. When P _max is <10 mN, the micro-hardness and the elastic modulus increase linearly with the increase of P _max. When P _max exceeds 10 mN, the micro-hardness decreases with the increase of P _max and then gradually tends to be stable, and the elastic modulus increases by power function with the increase of P _max and then gradually tends to be stable.     

Tribological behavior of B4C reinforced Fe-base bulk metallic glass composite coating

In the present study, the tribological behavior of B₄C reinforced Fe-based bulk metallic glass (BMG) in the form of spray coatings was investigated. These coatings were successfully deposited on mild steel substrates using shrouded plasma spray techniques. The B₄C fraction and distribution in the deposited BMG/B₄C coatings were evaluated by image analysis and scanning electron microscopy. Friction and wear experiments were performed under dry conditions using a pin-on-disk sliding wear test against SUJ2 countermaterial for different B₄C fractions. It was observed that the wear resistance of composite coatings was greatly improved relative to the BMG coating. The results show that the friction coefficient of BMG/B₄C coatings is dependent on the fraction of B₄C in the BMG matrix. The wear behavior of Fe-based BMG is governed by plastic deformation and fracture of the wear surface. By embedding a harder material, B₄C, in a comparatively soft matrix, the hardness of the wear surface can be increased, and plastic flow propagation is inhibited. Moreover, the lower friction coefficient of B₄C can lead to reductions in wear loss.     

Gradients in elastic modulus for improved contact-damage resistance. part ii: the silicon nitride–silicon carbide system

In an effort to create elastic-modulus (E) graded materials for contact-damage resistance—free of substantial amounts of glass—silicon nitride (Si₃N₄)-silicon carbide (SiC) graded materials were processed. The structure of these graded materials is such that Si₃N₄ (E=300 GPa) is at the contact surface and SiC (E=400 GPa) is in the interior, with a stepwise gradient in composition existing between the two over a depth of 1.6 mm. A pressureless, liquid-phase co-sintering method, in conjunction with a powder-layering technique, was used to achieve this structure. The liquid phase used was yttrium aluminum garnet (YAG). Under spherical indentation, cone-cracks did not form in the graded material, but some inelastic shear deformation was observed. Cone cracks formed in both the monolithic Si₃N₄ and the monolithic SiC end member materials under identical indentation conditions. Finite element analysis (FEA) of the stresses associated with indentation revealed that the maximum principal tensile stresses outside the Hertzian contact circle, which drive the classical cone-cracks, are reduced by approximately 12% in the graded material relative to the monolithic silicon nitride case. This reduction is significantly lower than what was calculated for the Si₃N₄-glass case (Part I), owing to the shallower, linear E-gradient over a 1.6 mm depth in Si₃N₄-SiC, as compared with the power-law, steeper E-gradient over 0.4 mm depth in the Si₃N₄-glass. It appears that, in addition to the E-gradient, the inelastic deformation contributes to the suppression of cone cracks in the Si₃N₄-SiC graded material. It is suggested that compressive residual stresses may be present in the Si₃N₄-SiC graded material, which are also likely to aid in the suppression of cone-cracks.     

Structural transformations and wear resistance of titanium nickelide under conditions of sliding friction at a cryogenic (−196°C) temperature

Structural transformations and tribological properties of a Ti_49.4Ni_50.6 alloy have been investigated at the liquid-nitrogen temperature. It has been shown that the alloy under study possesses the resistance to abrasive and adhesive wear smaller by a factor of 1.4–1.7 and the friction coefficient that is (to 1.7 times) higher, as compared to the austenitic steel 12Kh18N9. The only moderate tribological properties of the titanium nickelide are caused by an enhanced brittleness of this material under the conditions of friction-initiated severe plastic deformation. The enhanced low-temperature brittleness of the martensitic structure is seemingly explained by a low symmetry of the crystal lattice of the B19’ martensite, an atomically ordered state of this phase, and the formation of a brittle amorphous phase in the layer several microns thick near the friction surface of the alloy. The appearance of a continuous amorphous layer at the friction surface of the titanium nickelide is favored by the presence of the martensitic structure in the alloy, its stability under the friction conditions with respect to the reverse B19′ → B2 transformation, and a high intensity of the deformation processes occurring in the zone of friction contact. Below the amorphous layer, a mixed amorphous-crystal-line structure is located. The nanocrystallites are textured and range in size from a few to tens of nanometers. The formation of crystallites of the B2 phase in the amorphized layer appears to occur at the stage of warming of the alloy samples to room temperature. A similar amorphous-nanocrystalline structure arises near the abrasive-wear surface of the Ti_49.4Ni_50.6 alloy. It has been shown that the presence of a submicrocrystalline structure in the initial Ti_49.4Ni_50.6 alloy exerts no significant effect on the tribological properties and the character of structural transformations induced in the alloy by the frictional action.