Hardness-packing density scaling relations for cohesive-frictional porous materials

Recent progress in instrumented nanoindentation makes it possible today to test in situ phase properties and structures of porous materials that cannot be recapitulated ex situ in bulk form. But it requires a rigorous indentation analysis to translate indentation data into meaningful mechanical properties. This paper reports the development and implementation of a multi-scale indentation analysis based on limit analysis, for the assessment of strength properties of cohesive-frictional porous materials from hardness measurements. Based on the separation-of-scale condition, we implement an elliptical strength criterion which results from the nonlinear homogenization of the strength properties of the constituents (cohesion and friction), the porosity and the microstructure, into a computational yield design approach to indentation analysis. We identify the resulting upper bound problem as a second-order conical optimization problem, for which advanced optimization algorithms became recently available. The upper bound yield design solutions are benchmarked against solutions from comprehensive elastoplastic contact mechanics finite element solutions and lower bound solutions. Furthermore, from a detailed parameter study based on intensive computational simulations, we identify characteristic hardness-packing density scaling relations for cohesive-frictional porous materials. These scaling relations which are developed for two pore-morphologies, a matrix-pore morphology and a polycrystal (perfect disordered) morphology, are most suitable for the reverse analysis of the strength parameters of cohesive-frictional solids from indentation hardness measurements.     

Mechanics of indentation of plastically graded materials—II: Experiments on nanocrystalline alloys with grain size gradients

A systematic study of depth-sensing indentation was performed on nanocrystalline (nc) Ni-W alloys specially synthesized with controlled unidirectional gradients in plastic properties. A yield strength gradient and a roughly constant Young's modulus were achieved in the nc alloys, using electrodeposition techniques. The force vs. displacement response from instrumented indentation experiments matched very well with that predicted from the analysis of Part I of this paper. The experiments also revealed that the pile-up of the graded alloy around the indenter is noticeably higher than that for the two homogeneous reference alloys that constitute the bounding conditions for the graded material. These trends are also consistent with the predictions of the indentation analysis.     

Orientation effect on the Boussinesq indentation of a transversely isotropic piezoelectric material

Three-dimensional finite element analysis was used to study the effect of the angle between the loading direction and the axisymmetric direction on the indentation behavior of a transversely isotropic piezoelectric half-space by a cylindrical indenter of flat end. Two cases were considered in the analysis, which included (a) the indentation by an insulating indenter, and (b) the indentation by a conducting indenter. Both the indentation load and the indentation-induced potential were found to be proportional to the indentation depth. Using the simulation results and the analytical relationship for the indentation by a rigid, insulating indenter, semi-analytical relationships were developed between the indentation load and the indentation depth and between the indentation-induced potential on the indenter and the indentation depth for the conducting indenter, respectively. The proportionality between the indentation-induced potential and the indentation depth is only a function of the angle between the loading direction and the poling direction, independent of the type of indenters, which may be used to measure the relative direction of the loading axis to the axisymmetric axis of transversely piezoelectric materials from the indentation test.     

On the uniqueness and sensitivity of indentation testing of isotropic materials

Instrumented indentation is a popular technique to extract the material properties of small scale structures. The uniqueness and sensitivity to experimental errors determine the practical usefulness of such experiments. Here, a method to identify test techniques that minimizes sensitivity to experimental erros is in indentation experiments developed. The methods are based on considering “shape functions,” which are sets of functions that describe the force–displacement relationship obtained during the indentation test. The concept of condition number is used to investigate the relative reliability of various possible dual indentation techniques. Interestingly, it was found that many dual indentation techniques can be as unreliable as single indentation techniques. Sensitivity analyses were employed for further understanding of the uniqueness and sensitivity to experimental errors of indentation techniques. The advantage of the Monte Carlo approach over other procedures is established. Practical guidelines regarding the selection of shape functions of force–displacement relationship and geometric parameters, while carrying out indentation analysis are provided. The results suggest that indentation experiments need to be very accurate to extract reliable material properties.     

Formation of indentation cracks and origin of indentation size effect in cadmium tartrate pentahydrate single crystals

The load dependence of the Vickers microhardness on the as-grown (0 1 0) and (0 0 1), and cleaved (0 0 1) faces of cadmium tartrate pentahydrate (CTPH) single crystals has been investigated. The experimental results showed that, with an increase in the applied load, the microhardness of the as-grown (0 1 0) and (0 0 1) faces decreases, while that of the unheated and heated (0 0 1) cleavage faces decreases first up to a load of 2.5 N and then increases. Analysis of the experimental results revealed that: (1) radial crack length, indentation size and applied indentation load are mutually related, and these dependences related with fracture mechanics are the basis of Meyer’s empirical law, (2) with increasing indentation load, changes in the mechanism of development of indentation cracks from radial cracks to lateral cracks and surface chipping of the material, followed by predominantly surface chipping of the material are responsible for indentation size effect in CTPH crystals, (3) proportional specimen resistance model and Meyer’s law not only explain the indentation size effect but also can be used to determine load-independent hardness H^∗, and (4) there is no direct relationship between microhardness and fracture toughness of different CTPH samples, while the values of load-independent hardness H^∗, and brittleness indices β and B of CTPH crystals increase linearly with the Meyer constant A. Procedures are given to determine load-independent hardness H^∗ from the transition values of load and corresponding indentation size.     

Polymer indentation: Numerical analysis and comparison with a spherical cavity model

Three dimensional analyses of indentation of a polymer by a rigid indenter are carried out. The polymer is characterized by a finite strain elastic-viscoplastic constitutive relation and the calculations are carried out with a dynamic finite element program used to simulate quasi-static behavior. Two types of indenter are considered; a conical indenter and a pyramidal indenter. For each indenter type, different values of the sharpness of the indenter are considered and two rates of indentation are analyzed. Significant sink-in is found to occur in all cases considered. The amount of sink-in is found to be smaller for sharper indenters. The calculated values of both the nominal and true hardness do not differ significantly for the two indenter shapes. An expanding spherical cavity model is also considered and the predictions of this model are compared with the finite element results for various indenter shapes and indentation rates. The spherical cavity model is found to give fairly good agreement with the predictions of the finite element analyses for the nominal polymer hardness for both indenter shapes.     

Hybrid cyanate ester resin-based nanocomposites: Increased indentation size effect due to anomalous composition of micron subsurface layer

Hardness of materials increases with decreasing indentation depth from macro-to nano-scales, which is known as the indentation size effect (ISE). This effect has been associated with indenter shape, frictional forces, dislocation models and other features. We show an anomalously high ISE for a 1-μm subsurface layer in the hybrid nanocomposites based on densely cross-linked Cyanate Ester Resins (CER) containing functionalized 3-D POSS or 2-D MMT nanoparticles (NP). This effect disappears after mechanical stripping of the surface layer. Energy dispersive X-ray (EDX) spectral analysis shows that this anomaly was caused by increased content of NP (Si and Al elements), by 2.5 times, in the 1-μm subsurface layer. The hardness of the 1-μm subsurface layer in these brittle nanocomposites is due to its peculiar composition, and must be taken into account when considering mechanical strength and frictional properties.     

Material Property Differentiation in Indentation Testing Using Secondary Sensors

Current in vivo and in situ testing procedures are dominated by indentation. The major challenge for this testing technique is in finding a unique solution to the “inverse problem” i.e., defining an appropriate constitutive framework and obtaining material properties consistent with the indentation force–displacement data. Much of the information related to the interplay between shear and bulk compliance in the deformation field beneath the indenter is lost when capturing this single output. We propose a material testing method that follows the well proven path of conventional indentation methods, but enriches the signal by acquiring displacement data not only for the actuated indenter, but also for a set of offset, passive secondary sensors. We use finite element (FE) simulations involving three cases of materials: (a) linear elastic, (b) hyperelastic and (c) time-dependent to demonstrate the benefit of these additional sensors. The results indicate that the addition of these secondary sensors can help to discern between materials with varying degrees of compressibility.     

Wedge indentation into elastic-plastic single crystals, 1: Asymptotic fields for nearly-flat wedge

Asymptotic stress and deformation fields under the contact point singularities of a nearly-flat wedge indenter and of a flat punch are derived for elastic ideally-plastic single crystals with three effective in-plane slip systems that admit a plane strain deformation state. Face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal-close packed (HCP) crystals are considered. The asymptotic fields for the flat punch are analogous to those at the tip of a stationary crack, so a potential solution is that the deformation field consists entirely of angular constant stress plastic sectors separated by rays of plastic deformation across which stresses change discontinuously. The asymptotic fields for a nearly-flat wedge indenter are analogous to those of a quasistatically growing crack tip fields in that stress discontinuities can not exist across sector boundaries. Hence, the asymptotic fields under the contact point singularities of a nearly-flat wedge indenter are significantly different than those under a flat punch. A family of solutions is derived that consists entirely of elastically deforming angular sectors separated by rays of plastic deformation across which the stress state is continuous. Such a solution can be found for FCC and BCC crystals, but it is shown that the asymptotic fields for HCP crystals must include at least one angular constant stress plastic sector. The structure of such fields is important because they play a significant role in the establishment of the overall fields under a wedge indenter in a single crystal. Numerical simulations—discussed in detail in a companion paper—of the stress and deformation fields under the contact point singularity of a wedge indenter for a FCC crystal possess the salient features of the analytical solution.