Flat-ended Indenter Research Articles

Measurement of fracture toughness in high-strength alloys via modified limit load analysis using flat-end cylindrical indenter

Measurement of fracture toughness in high-strength alloys via modified limit load analysis using flat-end cylindrical indenter

A porous layer of negative value of Poisson’s ratio under a flat-ended and rigid cylinder indenter

A porous layer of negative value of Poisson’s ratio under a flat-ended and rigid cylinder indenter

Influence of surface and couple stresses on response of elastic substrate under tilted flat indenter

The simultaneous influence of surface and couple stresses on the nonsymmetrical frictionless indentation of a linearly elastic, homogenous, and isotropic half plane under a tilted, rigid, flat-ended indenter with sharp, square corners was investigated by adopting existing continuum-based models. The half plane was mathematically constituted by the surface layer and bulk materials, and the mechanical response of each component was described by means of the Gurtin-Murdoch surface-elasticity and couple-stress elasticity theories, respectively. Under an applied eccentric load, the geometric compatibility condition for points within the contact region and the equilibrium conditions of the indenter were utilized to establish a set of governing equations for the unknown contact pressure. Selected efficient numerical schemes, based mainly upon the solution discretization, the standard collocation technique, and the fundamental solution of an elastic half plane under surface normal loads, were developed to construct the solution of complete and receding contacts numerically. Obtained results from an extensive numerical study apparently reflect the crucial effects of the surface and couple stresses on the indentation behaviors of the tilted indenters and the elastic response of the bulk material, and, in particular, the strong size-dependency of solutions when the width of the indenter becomes comparable to the length scale parameter of the surface and bulk materials. These characteristics not only provide valuable information for a better understanding of the mechanical properties of materials by micro-/nano-indentation tests, but also facilitate a basis for further investigations in the area of micro-/nano-mechanics.

On the receding contact between a graded and a homogeneous layer due to a flat-ended indenter

This paper solves the frictionless receding contact problem between a graded and a homogeneous elastic layer due to a flat-ended rigid indenter. Although its Poisson’s ratio is kept as a constant, the shear modulus in the graded layer is assumed to exponentially vary along the thickness direction. The primary goal of this study is to investigate the functional dependence of both contact pressures and the extent of receding contact on the mechanical and geometric properties. For verification and validation purposes, both theoretical analysis and finite element modelings are conducted. In the analytical formulation, governing equations and boundary conditions of the double contact problem are converted into dual singular integral equations of Cauchy type with the help of Fourier integral transforms. In view of the drastically different singularity behavior of the stationary and receding contact pressures, Gauss–Chebyshev quadratures and collocations of both the first and the second kinds have to be jointly used to transform the dual singular integral equations into an algebraic system. As the resultant algebraic equations are nonlinear with respect to the extent of receding contact, an iterative algorithm based on the method of steepest descent is further developed. The semianalytical results are extensively verified and validated with those obtained from the graded finite element method, whose implementation details are also given for easy reference. Results from both approaches reveal that the property gradation, indenter width, and thickness ratio all play significant roles in the determination of both contact pressures and the receding contact extent. An appropriate combination of these parameters is able to tailor the double contact properties as desired.

The effect of highly inhomogeneous biphasic properties on mechanical behaviour of articular cartilage

Background and objectiveInvestigating the biomechanics of cartilage could help to understand the unique load-bearing property of the cartilage and optimize the scaffold design in tissue-engineering. It is important to model the cartilage as a highly inhomogeneous fibril-reinforced biphasic material to represent its complex composition and structure. The depth-dependent and strain-dependent properties of the cartilage would also play an important role in its mechanical behaviour. However, the differences in representing the cartilage as a highly inhomogeneous model or as simplified models still remain unclear. Hence, in this study, a highly inhomogeneous fibril-reinforced biphasic cartilage model considering both the depth-dependent and strain-dependent properties was constructed; the effect of highly inhomogeneous properties on the mechanical behaviour of articular cartilage was investigated. MethodsA finite element model of the cartilage was developed based on a flat-ended indentation test. Compressive forces were applied to four various inhomogeneous layered models through a porous indenter (Model 1: nine layers with strain-dependent permeability; Model 2: three layers with strain-dependent permeability; Model 3: single layer with strain-dependent permeability; Model 4: nine layers with constant permeability). ResultsModels 1 and 2 provided similar results with less than 3% difference in the peak effective stress, contact pressure, fluid pressure as well as fluid support ratio. However, Model 1 to Model 3 differed in stress and strain distribution patterns along depth over prolonged loads, which may provide an important insight into the highly inhomogeneous depth-dependent properties of cartilage. In addition, Model 1 with strain-dependent permeability demonstrated an enhanced capability on fluid pressurisation as compared with Model 4 which had constant permeability. ConclusionsA highly inhomogeneous fibril-reinforced biphasic model considering both depth-dependent and strain-dependent properties was developed in this study, in order to illustrate the effect of highly inhomogeneous properties on the mechanical behaviour of the articular cartilage. The number of layers in the models with depth-dependent properties should be selected according to the research questions and clinical demands. The model with strain-dependent permeability offers an enhanced capability on fluid pressurisation. In future studies, the proposed model could be adopted in cell-models to provide more in-depth information or in tissue-engineering to optimize the depth-dependent scaffold structure.

Experimental investigation of quasi-static behavior of composite and fiber metal laminate panels modified by graphene nanoplatelets

The current study investigated the influence of incorporation of graphene nanoplatelets (GNPs) on quasi-static behavior of composite and fiber metal laminate (FML) panels. The unmodified and modified composite specimens and FML panels with 2/1 configuration were fabricated using a hand lay-up method and investigated through a quasi-static punch and indentation testing. The two sets of tests were conducted with a flat-ended indenter and the loading conditions were the same for all samples, except support spans which were varied. Following experimental testing, possible damages at the punch region were closely investigated and localized and global damages were observed. The results revealed that adding 0.2 wt% GNPs improved the strength and fracture toughness of specimens by delaying the failure modes. On the contrary, GNPs made the bonding between the aluminum and composite interface to weaken.

Contact-induced change of the bandgap of semiconductors of the wurtzite structure

Progress in the synthesis of semiconductors of nanoscale structures has revived an interest in using semiconductors of different structures in the sensing of mechanical deformation through the change of photoluminescent characteristics under external stress/strain. In the heart of the deformation-induced change of photoluminescent characteristics is the deformation-induced change of the bandgap of semiconductors. In this work, we investigate the bandgap changes of bulk semiconductors of the wurtzite structure due to elastic indentation by a rigid, axisymmetric indenter. Using deformation potentials and the analytical solutions for the contact problems, we obtain closed-form solutions of the contact-induced change of the bandgaps for the elastic indentation by three different indenters of flat-ended, spherical, and conical shapes. For the elastic indentation by a flat-ended indenter, the bandgap changes are proportional to the indenter displacement/indentation load. For the elastic indentation by a spherical indenter, the bandgap changes are proportional to the cubic root of the indention load. For the elastic indentation by a conical indenter, the bandgap changes exhibit the characteristic of shape-similarity, independent of the indentation depth/load.

Ferroelastic and plastic behaviors in pseudo-single crystal micropillars of nontransformable tetragonal zirconia

The orientation-dependent micromechanical properties of nontransformable tetragonal (t’) zirconia, which underwent a diffusionless transformation from the fluorite cubic phase and does not exhibit a stress-induced phase transformation, were characterized via pseudo-single crystal micropillar compression and electron microscopy. The t’ zirconia sample was obtained via atmospheric plasma spraying of 4.5 mol% yttria-stabilized zirconia (YSZ) powders into liquid nitrogen and consolidated into a bulk state via hot pressing at 1100°C. Dense and cylindrical micropillars were fabricated using a focused ion beam from pseudo-single crystalline regions, which exhibited a nanodomain microstructure of three t’ variants partitioned by {1 0 1}c twin boundaries with 90° symmetry. These micropillars were compressed using a flat-end diamond indenter. Near-<0 0 1>c compressions were attributed to ferroelastic domain switching and subsequent {1 0 1}c and/or {1 1 1}c hard slips. In ferroelastic deformation, a certain t’ variant diminished, and a binary domain microstructure developed with c axes perpendicular to the compressive direction. Near-<1 1 1>c compressions were governed by {0 0 1}c soft slips accompanied by strain hardening with negligible ferroelasticity, which resulted in buckling deformation with rotational kinking. In both the hard- and soft-slip orientations, ferroelastic toughening was observed with certain t’ variants awaken around the crack tips. Contrarily, cleavage fractures subsequent to yielding were observed in near-<1 0 1>c compressions. In the cubic counterpart with a domain-free microstructure (8.0 mol% YSZ), ferroelastic toughening was not observed. Hence, it is viewed as the origin of enhanced toughness in t’ zirconia.

Ferroelastic and Plastic Behaviors in Pseudo-Single Crystal Micropillars of Nontransformable Tetragonal Zirconia

The orientation-dependent micromechanical properties of nontransformable tetragonal (t') zirconia, which diffusionlessly transformed from the fluorite cubic phase and does not show stress-induced phase transformation, were characterized via pseudo-single crystal micropillar compression and electron microscopy. The t' zirconia sample was obtained via atmospheric plasma spraying of 4.5 mol% yttria-stabilized zirconia (YSZ) powders into liquid nitrogen and consolidated into a bulk state via hot pressing at 1100°C. Dense and cylindrical micropillars were fabricated using a focused ion beam from pseudo-single crystalline regions, which had a nanodomain microstructure of three t' variants partitioned by {1 0 1}c twin boundaries with 90° symmetry, and were compressed using a flat-end diamond indenter. Near- c compressions were attributed to ferroelastic domain switching and subsequent {1 0 1}c and/or {1 1 1}c slips. In ferroelastic deformation, a certain t' variant diminished and a binary domain microstructure developed with c axes perpendicular to the compressive direction. Near- c compressions were governed by {0 0 1}c slips accompanied by strain hardening with negligible ferroelasticity, resulting in buckling deformation with rotational kinking. Crack-tolerant plasticity was ascribed to ferroelastic toughening, where t' variants with c axes across crack planes developed to relieve stress concentrations around the crack tips in both orientations. In contrast, pseudo-cleavage fractures on low-index planes were observed in near- c compressions. Crack-tolerant behavior was not observed in the cubic counterpart with a domain-free microstructure (8.0 mol% YSZ), which demonstrated catastrophic fractures. Hence, ferroelastic toughening is viewed as the origin of enhanced toughness in t' zirconia.

Physical modeling and arbitrary Lagrangian\u2013Eulerian finite element analysis of indentation of a sensitive clay by a flat-ended axisymmetrical indenter

Because indentation tests which are used in several engineering fields allow obtaining material strength parameters in a fast, reliable and reproducible manner, the present study was initiated with the aim of finding out whether such tests could also be used in geotechnical engineering. Experimental results are obtained from undrained indentation tests performed with a rigid flat-tipped, cylindrical indenter on Champlain clay specimens in order to deduce values of elastic moduli and yield stresses. These values are compared to those deduced from unconfined compression tests. Results of an Arbitrary Lagrangian–Eulerian (ALE) based finite element analysis that simulates such test are also presented. It is assumed that the clay behaves as a linear-elastic, perfectly plastic material obeying von-Mises yield criterion. A frictionless contact interface is chosen to realistically model interactions on the indenter-clay, clay-platen and clay-ring interfaces. The paper presents distributions of contact pressures beneath the indenter and along lateral and lower boundaries. Typical contours of von Mises deviator stress and equivalent plastic strain corresponding to different indentation depths in the clay specimen are presented. It is also shown that the presence of hairline cracks on the clay along the perimeter of the indenter and the limited thickness of the sample affect the deduced values of Young’s modulus and yield stress.

Neural network implementation for the prediction of load curves of a flat head indenter on hot aluminum alloy

The indentation test performed by means of a flat-ended indenter is a valuable non-destructive method for assessment of metals at a local scale. Particularly, from the indentation curves it is possible to achieve several mechanical properties. The aim of this paper is the implementation of an artificial neural network for the prediction of the indentation load as a function of the penetration depth for an aluminium substrate. In particular, the neural network is addressed to the mechanical characterization of the bulk in function of temperature and indentation rate. The results obtained showed a high accuracy in curves prediction.

Surface Zener–Stroh crack model to slip band due to contact

A surface slip band due to contact by a rectangular rigid flat-ended indenter is investigated. An inclined Zener–Stroh crack model is proposed to simulate the slip band. By using the fundamental solution of a single dislocation in a half plane as Green’s function, the Zener–Stroh crack is modeled with continuously distributed dislocations. It leads to a singular integral equation of the first kind, which is solved with the Gauss–Chebyshev numerical quadrature, and then stress intensity factors (SIFs) at the crack tips are evaluated. It is demonstrated that the Zener–Stroh crack model can efficiently capture micro deformation behavior of the surface slip band due to contact. With this model, the corresponding relations of the applied load, the slip band length, the relative sliding displacement of slip band and SIFs are obtained. Compared with the experimental results, it is shown that the surface Zener–Stroh crack model to contact slip band can well address such kind of contact damage problems.

Determining Prestrains in an Elastomer Through Elliptical Indentation

Residual stresses or prestrains could strongly affect the properties and functionalities of soft materials and tissues. However, non-destructive measurements of residual stresses or prestrains in materials are generally challenging. Previous studies have shown that residual stresses or prestrains can affect indentation tests of different materials, including metals, glassy polymers, soft elastomers, and gels. Nevertheless, an indentation method for determining the state of residual stresses or prestrains of large magnitude in an elastomer is not yet available. In this article, we propose to use elliptical indentation to measure large prestretches in a Neo-Hookean elastomer. We have analytically derived the relationship between the indentation force and indentation depth for both a flat-ended elliptical indenter and an ellipsoidal indenter. We have further shown that such a relationship greatly depends on the rotational angle of the indenter with respect to the direction of principle stretches in the elastomer. Based on the derived analytical results, we construct two diagrams for the flat-ended elliptical indenter and ellipsoidal indenter, respectively, which can be directly used to determine quantitatively large prestrains in an elastomer.

Adhesive contact between a rigid body of arbitrary shape and a thin elastic coating

Application of the principle of energy balance to a rigid indenter in contact with a thin elastic layer on a flat rigid substrate provides a very simple derivation of the detachment criterion which earlier has been obtained by much more complicated asymptotic analysis. The simple criterion is additionally confirmed by the fully three-dimensional simulations of contact with a coated rigid substrate using the recently developed formulation of the boundary element method for coated media. The found detachment criterion is applied to contact of indenters of various shape. In the case of flat-ended indenters, the adhesive strength occurs to be proportional to the area of the face of the indenter (independently of the shape). The asymptotic criterion is also used for calculation of the adhesion strength of indenters having arbitrary shape and is illustrated with a case study of a contact of a rough indenter with a coated substrate.

Models for evaluating craters morphology, relation of indentation hardness and uniaxial compressive strength via a flat-end indenter

AbstractIn this work, the intensive theoretical study and laboratory tests are conducted to evaluate the craters morphology via the flat-ended indenter test, relationship of indentation hardness (HRI) and uniaxial compressive strength (UCS). Based on the stress distribution, failure process and Mohr–Coulomb failure criterion, the mathematical mechanical models are presented to express the formation conditions of “pulverized zone” and “volume break”. Moreover, a set of equations relating the depth and apex angle of craters, the ratio of indentation hardness and uniaxial compressive strength, the angle of internal friction and Poisson’s ratio are obtained. The depth, apex angle of craters and ratio of indentation hardness and uniaxial compressive strength are all affected by the angle of internal friction and Poisson’s ratio. The proposed models are also verified by experiments of rock samples which are cored from Da Qing oilfield, the percentage error between the test and calculated results for depth, apex angle of craters and the ratio of HRI and UCS are mainly in the range of –1.41%–8.92%, –5.91%–3.94% and –8.22%–13.22% respectively for siltstone, volcanic tuff, volcanic breccia, shale, sand stone and glutenite except mudstone, which demonstrates that our proposed models are robust and effective for brittle rock.

Analytic relations for two-dimensional indentations with surface tension

Surface tension plays a significant role in micro- and nanoindentation tests. Based on the solution of a concentrated force acting on an elastic half-plane with surface tension, the two-dimensional indentations of an elastic half-plane by a cylindrical indenter, a wedge indenter and a flat-ended indenter are formulated, and by employing the Gauss–Chebyshev quadrature formula, the corresponding singular integral equations are numerically solved. For each indentation, the analytic solutions of two limit cases considering only bulk elasticity or surface tension are derived. Then through simple combinations of the results of two limit cases, the analytic relations between load and contact half-width or indent depth are formulated for the general case, and compared with the numerical results. The analytic relations established in this paper facilitate better characterization of mechanical properties of materials by micro/nanoindentation tests.

Indentation theory on a half-space of transversely isotropic multi-ferroic composite medium: sliding friction effect

This article aims to present a systematic indentation theory on a half-space of multi-ferroic composite medium with transverse isotropy. The effect of sliding friction between the indenter and substrate is taken into account. The cylindrical flat-ended indenter is assumed to be electrically/magnetically conducting or insulating, which leads to four sets of mixed boundary-value problems. The indentation forces in the normal and tangential directions are related to the Coulomb friction law. For each case, the integral equations governing the contact behavior are developed by means of the generalized method of potential theory, and the corresponding coupling field is obtained in terms of elementary functions. The effect of sliding on the contact behavior is investigated. Finite element method (FEM) in the context of magneto-electro-elasticity is developed to discuss the validity of the analytical solutions. The obtained analytical solutions may serve as benchmarks to various simplified analyses and numerical codes and as a guide for future experimental studies.

Mapping of Two-Dimensional Contact Problems on a Problem with a One-Dimensional Parametrization

We discuss a possible generalization of the ideas of the method of dimensionality reduction (MDR) for the mapping of two-dimensional contact problems (line contacts). The conventional formulation of the MDR is based on the existence and uniqueness of a relation between indentation depth and contact radius. In two-dimensional contact problems, the indentation depth is not defined unambiguously, thus another parametrization is needed. We show here that the Mossakovskii-Jager procedure of representing a contact as a series of incremental indentations by flat-ended indenters can be carried out in two-dimensions as well. The only available parameter of this process is, however, the normal load (instead of indentation depth as in the case of threedimensional contacts). Using this idea, a complete solution is obtained for arbitrary symmetric two-dimensional contacts with a compact contact area. The solution includes both the relations of force and half-width of the contact and the stress distribution in the contact area. The procedure is generalized for adhesive contacts and is illustrated by solutions of a series of contact problems.

Nonlinear compliance of elastic layers to indentation.

Thin samples adherent to a rigid substrate are considerably less compliant to indentation when compared to specimens that are not geometrically confined. Analytical corrections to this so-called substrate effect exist for various types of indenters but are not applicable when large deformations are possible, as is the case in biological materials. To overcome this limitation, we construct a nonlinear scaling model characterized by one single exponent, which we explore employing a parametric finite element analysis. The model is based on asymptotes of two length scales in relation to the sample thickness, i.e., indentation depth and radius of the contact area. For small indentation depth, we require agreement with analytical, linear models, whereas for large indentation depth and extensive contact area, we recognize similarity to uniaxial deformation, indicating a divergent force required to indent nonlinear materials. In contrast, we find linear materials not to be influenced by the substrate effect beyond first order, implying that nonlinear effects originating from either the material or geometric confinement are clearly separated only in thin samples. Furthermore, in this regime the scaling model can be derived by following a heuristic argument extending a linear model to large indentation depths. Lastly, in a large indentation setting where the contact is small in comparison with sample thickness, we observe nonlinear effects independent of material type that we attribute to a higher-order influence of geometrical confinement. In this regime, we define a scalar as the ratio of strains along principal axes as obtained by comparison with the case of a point force on a half-space. We find this scalar to be in quantitative agreement with the scaling exponent, indicating an approach to distinguish between nonlinear effects in the scaling model. While we conjecture our findings to be applicable to other flat-ended indenters, we focus on the case of a flat-ended cylinder in normal contact with a thin layer. The analytical solution for small indentation associated with this problem has been given by Hayes et al. (J Biomech 5:541-551, 1972), for which we provide a convenient numerical implementation.

Material property analytical relations for the case of an AFM probe tapping a viscoelastic surface containing multiple characteristic times.

We explore the contact problem of a flat-end indenter penetrating intermittently a generalized viscoelastic surface, containing multiple characteristic times. This problem is especially relevant for nanoprobing of viscoelastic surfaces with the highly popular tapping-mode AFM imaging technique. By focusing on the material perspective and employing a rigorous rheological approach, we deliver analytical closed-form solutions that provide physical insight into the viscoelastic sources of repulsive forces, tip–sample dissipation and virial of the interaction. We also offer a systematic comparison to the well-established standard harmonic excitation, which is the case relevant for dynamic mechanical analysis (DMA) and for AFM techniques where tip–sample sinusoidal interaction is permanent. This comparison highlights the substantial complexity added by the intermittent-contact nature of the interaction, which precludes the derivation of straightforward equations as is the case for the well-known harmonic excitations. The derivations offered have been thoroughly validated through numerical simulations. Despite the complexities inherent to the intermittent-contact nature of the technique, the analytical findings highlight the potential feasibility of extracting meaningful viscoelastic properties with this imaging method.