Indenter Tip Research Articles

Sliding friction on particle filled epoxy: Developing a quantitative model for complex coatings

Epoxy resins represent an important class of thermosetting polymers that are extensively used in demanding applications like in scratch resistant coatings. Usually fillers, either hard (inorganic) or soft (rubbery), are added. Here we test hard and soft particle-filled epoxy systems in single asperity sliding friction experiments, and analyze the results with the hybrid numerical-experimental approach presented earlier. The focus is on the detailed modeling of the local deformation processes and it is confirmed that a rate-independent friction model proves appropriate to quantitatively model this complex process. The constitutive framework developed for amorphous thermoplastic polymers adequately describes also these thermoset systems. The materials response during scratching is likewise. Hard fillers decrease the penetration of the indenter into the surface, and consequently enhance scratch resistance; they cause the lateral friction force to decrease, since less material flows in front of the indenter tip. Soft fillers increase the penetration into the surface, according to expectations, but surprisingly also decrease the friction force.Simulations do not predict this, and suggest an alternative explanation. Migration of rubber particles during sample preparation to the surface could have occurred. Adding a thin rubbery layer to the surface makes the model quantitative, but SEM and TEM pictures of the cross-sections do not confirm this phase separation and instead show the presence of a large number of very small voids. Including these voids in the modeling allows to predict the penetration depth into the surface and lateral force quantitative for all sliding speeds.

Nanomechanical Strain Concentration on a Two-Dimensional Nanobridge within a Large Suspended Bilayer Graphene for Molecular Mass Detection

The recent emergence of strain gradient engineering directly affects the nanomechanics, optoelectronics and thermal transport fields in 2D materials. More specifically, large suspended graphene under very high stress represents the quintessence for nanomechanical mass detection through unique molecular reactions. Different techniques have been used to induce strain in 2D materials, for instance by applying tip indentation, pressure or substrate bending on a graphene membrane. Nevertheless, an efficient way to control the strain of a structure is to engineer the system geometry as shown in everyday life in architecture and acoustics. Similarly, we studied the concentration of strain in artificial nanoconstrictions (~100 nm) in a suspended epitaxial bilayer graphene membrane with different geometries and lengths ranging from 10 to 40 micrometer. We carefully isolated the strain signature from micro-Raman measurements and extracted information on a scale below the laser spot size by analyzing the broadened shape of our Raman peaks, up to 100 cm-1. We potentially measured a strong strain concentration in a nanoconstriction up to 5percent, which is 20 times larger than the native epitaxial graphene strain. Moreover, with a bilayer graphene, our configuration naturally enhanced the native asymmetric strain between the upper and lower graphene layers. In contrast to previous results, we can achieve any kind of complex strain tensor in graphene thanks to our structural approach. This method completes the previous strain-induced techniques and opens up new perspectives for bilayer graphene and 2D heterostructures based devices.

Chemical controls on the propagation rate of fracture in calcite

Calcite (CaCO3) is one of the most abundant minerals in the Earth’s crust, and it is susceptible to subcritical chemically-driven fracturing. Understanding chemical processes at individual fracture tips, and how they control the development of fractures and fracture networks in the subsurface, is critical for carbon and nuclear waste storage, resource extraction, and predicting earthquakes. Chemical processes controlling subcritical fracture in calcite are poorly understood. We demonstrate a novel approach to quantify the coupled chemical-mechanical effects on subcritical fracture. The calcite surface was indented using a Vickers-geometry indenter tip, which resulted in repeatable micron-scale fractures propagating from the indent. Individual indented samples were submerged in an array of aqueous fluids and an optical microscope was used to track the fracture growth in situ. The fracture propagation rate varied from 1.6 × 10−8 m s−1 to 2.4 × 10−10 m s−1. The rate depended on the type of aqueous ligand present, and did not correlate with the measured dissolution rate of calcite or trends in zeta-potential. We postulate that chemical complexation at the fracture tip in calcite controls the growth of subcritical fracture. Previous studies indirectly pointed to the zeta-potential being the most critical factor, while our work indicates that variation in the zeta-potential has a secondary effect.

On the local mechanical properties of isothermally transformed bainite in low carbon steel

For bainite microstructure composed of ferritic matrix and carbides or martensite-austenite (MA) constituents, one of constituent phases sometimes plays a crucial role in the specific stage of overall mechanical behaviors. In this study, the nanoindentation tests were employed to examine the local mechanical behavior of each constituent phase that formed after isothermal annealing. Transmission electron microscopy observation was conducted to reveal the intricate structure of each constituent phase. The average nanohardness of MA constituents decreased sharply with increasing isothermal temperature, while the ferritic laths had relatively stable nanohardness over the whole range of bainite transformation. However, the nanohardness of ferritic laths was intrinsically lower than the nanohardness of bainite reported in the literature. For the bainite formed at lower isothermal temperature condition, the MA constituents exhibited twice the hardness of ferritic matrix. Based on the load-displacement curve, it was further revealed that the MA constituents have higher the maximum shear stress and smaller plastic zone underneath the indenter tip than the ferritic matrix, leading to the difference in plastic deformation ability between them.

Comment on van Casteren et al. (2018): softer metallic spheres do abrade harder enamel.

Comment on van Casteren et al. (2018): softer metallic spheres do abrade harder enamel.

Numerical Simulation of the Depth-Sensing Indentation Test with Knoop Indenter

Depth-sensing indentation (DSI) technique allows easy and reliable determination of two mechanical properties of materials: hardness and Young’s modulus. Most of the studies are focusing on the Vickers, Berkovich, and conical indenter geometries. In case of Knoop indenter, the existing experimental and numerical studies are scarce. The goal of the current study is to contribute for the understanding of the mechanical phenomena that occur in the material under Knoop indention, enhancing and facilitating the analysis of its results obtained in DSI tests. For this purpose, a finite element code, DD3IMP, was used to numerically simulate the Knoop indentation test. A finite element mesh was developed and optimized in order to attain accurate values of the mechanical properties. Also, a careful modeling of the Knoop indenter was performed to take into account the geometry and size of the imperfection (offset) of the indenter tip, as in real cases.

Assessment of yield strength mismatch in X80 pipeline steel welds using instrumented indentation

Strain-based design (SBD) of pipelines allows stress owing to displacement-controlled loads originating from landslides, seismic motions, or frost heaves, to exceed yield stress. In such cases, the distribution of the strain over a large area (pipe) rather than local area (weld) is preferred. Accordingly, in SBD, yield strength of the circumferential girth weld is over matched when compared to the base metal. In this work, API-X80 pipeline steel sections were joined together using the robotic Gas-Metal Arc Welding (GMAW) process; filler metals and shielding gases were varied to achieve three different levels of strength (even, over and under match) between the weld and base metal. Weld metal and HAZ cross-section microstructures were investigated and correlated with micro-hardness maps to ensure that different mismatch levels were achieved. Yield strengths of the welds in the hoop direction were measured using both instrumented indentation and conventional tensile testing. The instrumented indentation technique used a 100-μm diameter nearly-flat indenter to estimate yield strengths for the welds and heat-affected zones from the load-displacement response for each zone. The measured yield strengths were compared with 0.2% offset tensile yield strength obtained through conventional tensile testing of all-weld and base metal specimens. Strength measurements, in the hoop direction, for weld metal from both of the techniques agreed to within 4.6%. The results showed that the nearly-flat tip indentation method can be used as a tool to directly estimate strength mismatch of pipeline girth welds. Measurements of yield strength for narrow coarse and fine grained heat-affected zones, which cannot be measured through any other method, can also be assessed using the techniques described herein.

Modeling of Osteoprobe indentation on bone

Osteoprobe (ActiveLife, Santa Barbara, CA) is a novel handheld microindentation instrument designed to test bone in vivo by measuring a Bone Material Strength index (BMSi). In this paper, the Osteoprobe indentation on a cortical bone is modeled computationally to gain insights into the physical interpretation of the BMSi output. The analysis is conducted using an axisymmetric finite element model with an isotropic viscoelastic-plastic constitutive law with continuum damage. The computational model is validated by comparing it with experimental data from the literature. Experimental factors (indenter tip radius and friction coefficient between the indenter and the bone) and four mechanical properties of bone (Young's modulus, compressive yield stress, and damage and viscosity constants) are varied to study their influence on the BMSi.We find that varying the friction coefficient can proportionally change the BMSi up to 3%. The indenter tip radius is proportional to the BMSi, with a more pronounced proportional relation when it is greater than 30 µm. Young's modulus has a proportional relation with the BMSi, where decreasing it by 73% reduces the BMSi by 41%. The damage constant has an inversely proportional relation to the BMSi, where increasing it from 0.5 to 0.96 reduces the BMSi by 29%. The compressive yield stress and the viscosity constant have a close proportional relation to the BMSi, where increasing the compressive yield stress from 50 MPa to 200 MPa increases the Osteoprobe BMSi by 21%.In summary, the friction coefficient and the indenter tip radius (when smaller than 30 µm) have a small effect on BMSi. Young's modulus and damage have stronger relations with the BMSi than compressive yield stress and viscosity constant. This fundamental study provides new insights into the BMSi measurement and serves as a basis for further computational and experimental investigations on the Osteoprobe technique. Such research is needed to facilitate the embrace of this technique by the clinical community.

Analysis of indentation measured mechanical properties on Multilayer Ceramic Capacitors (MLCCs)

Most failures of Multilayer Ceramic Capacitors (MLCCs) are mechanical (e.g., cracking) in nature. Therefore, reliability assessment of these parts depends heavily on the available mechanical properties in the literature. As indentation offers advantages such as minimal sample preparation and access to small areas, more mechanical properties on MLCCs based on measurement by indentation are being published in literature. However, indentation measured properties are known to vary with different factors such as measuring loads, geometry of indenter tip, and residual stresses. In this article, indentations are performed on commercial MLCCs at different loads and locations with different indenter tips. By identifying the causes of variations and understanding the limitations and implications of the indentation measured properties, the use of indentation for reliability assessment of MLCCs, and other electronic components can be facilitated.

Deformation and fracture in (0001) and (10-10) GaN single crystals

ABSTRACTIndentation techniques were utilised to induce deformation on polar (0001) c-plane and non-polar m-plane GaN single crystal. Cracking was more sensitively dependent on the orientation of the indenter tip, compared to hardness. The indentation-induced plastic deformation and fracture sequences were studied by cathodoluminescence imaging and optical microscopy, respectively. Polar GaN was harder than non-polar, while pop-in discontinuities occurred at lower loads in polar than non-polar GaN. Dislocation arrangements were more isotropic at the polar than the non-polar orientation. Polar GaN was more susceptible to cracking compared to non-polar. Indentation at the high load regime fostered radial and lateral crack formation at both indenter orientations in polar GaN. Post-indentation lateral crack propagation was observed in situ in polar GaN.This is part of a thematic issue on Nanoscale Materials Characterisation and Modeling by Advances Microscopy Methods - EUROMAT.

An acoustic emission sensor system for thin layer crack detection

We present a new method for semiconductor layer characterization enabling simultaneous crack generation and detection based on the established acoustic emission testing. In this paper, we explain the test concept and failure mechanism of diamond tip indentation with acoustic wave measurement on oxide layers by analytical modeling and computer-aided simulations. For this purpose, an innovative design of a sensor-indenter system was developed and experimentally verified. In further tests, the new acoustic crack detection method is correlated with proven optical inspection methods and the good reproducibility of the measurement is demonstrated. The high sensitivity and resolution of this method offer new opportunities for thin layer characterization at lower time and cost. The new high-efficient method can be used especially for process qualification for semiconductor wafer test, where oxide cracks can be induced by sharp probes causing electrical failure of the chip.

Inverse identification of constitutive parameters with instrumented indentation test considering the normalized loading and unloading P-h curves

The feature of instrumented indentation test (IIT) makes it suitable for the characterization of local constitutive behavior in a small volume of solids. In the present study, the relationship between the shape of the normalized loading P-h curve and the indenter tip rounding radius, and the influences of the constitutive parameters of a Ludwik-type isotropic hardening model on the shape and the initial slope of the normalized unloading P-h curve were investigated by theoretical analysis and sophisticated numerical sensitivity analysis. From the analytical results, a two-stage analytical approach that can inversely identify multiconstitutive parameters simultaneously was proposed. In the first stage, by considering the least-square distance between the experimental and the simulated normalized loading P-h curves (g(h/hmax)) as the objective and C1 in the tip area function as the input variable, the tip rounding was identified by a single-objective optimization procedure. With the optimal tip rounding, a multiobjective optimization procedure was then performed to identify the constitutive parameters of the target materials. The initial slope of the normalized unloading P-h curve (ΔS’(P/Pmax-h/hmax)), absolute value of the differences between the simulated and the experimental maximum load (ΔPmax), initial slope of the unloading P-h curve (ΔS), and contact hardness (ΔH) were set as objectives, while all the material parameters except the Poisson's ratio (v) were considered as input variables. Results of the numerical verification with a set of arbitrarily selected constitutive parameters and the experimental verification on Ti-18 (a near-β high-strength titanium alloy) fit fairly well with uniaxial tensile parameters. With regard to T40, the identified true stress-true plastic strain curve is about 20% higher than the uniaxial tensile curve. One possible explanation is the occurrence of the strength differential and anisotropy effects during the plastic deformation process of α-phase titanium. It is worthy to be noticed that the present study may propose a new way to calibrate the tip area function directly on the target material without any standard indentation sample and improve the quality of the identification of the material constitutive parameters by IIT.

Nanoscale Reciprocating Sliding Contacts of Textured Surfaces: Influence of Structure Parameters and Indentation Depth

Textured surfaces are widely used in engineering components as they can improve tribological properties of sliding contacts, while the detailed behaviors of nanoscale reciprocating sliding contacts of textured surfaces are still lack of study. By using multiscale method, two dimensional nanoscale reciprocating sliding contacts of textured surfaces are investigated. The influence of indentation depth, texture shape, texture spacing, and tip radius on the average friction forces and the running-in stages is studied. The results show that the lowest indentation depth can make all the four textured surfaces reach steady state. Surfaces with right-angled trapezoid textures on the right side are better for reducing the running-in stage, and surfaces with right-angled trapezoid textures on the left side are better to reduce wear. Compared with other textured surfaces, the total average friction forces can be reduced by 82.94%–91.49% for the case of the contact between the tip with radius R = 60r0 and the isosceles trapezoid textured surface. Besides, the total average friction forces increase with the tip radii due to that bigger tip will induce higher contact areas. This research proposes a detailed study on nanoscale reciprocating sliding contacts of textured surfaces, to contribute to design textured surfaces, reduce friction and wear.

Strain engineering in two-dimensional nanomaterials beyond graphene

Abstract Strain engineering is the process of tuning a material's properties by altering its mechanical or structural attributes. Atomically thin two-dimensional nanomaterials (2DNMs), which have been extensively studied in recent years, are particularly well-suited for strain engineering because they can withstand large strain. Thermal vibration, surface adhesion, substrate deformation, pre-stretched substrate, epitaxial grown, thermal expansion mismatch, substrate topography modification, pressurized blisters and tip indentation can lead to strain in 2DNMs. Strain in 2DNMs can modify their atomic structure, lattice vibration, thermal conductivity, electronic and optical, electrical and device performance, and chemical activities. This review focuses on the structural and mechanical properties of various 2DNMs, different experimental strategies to induce strain and modify properties, and applications of strained 2DNMs. Also, the review proposes prospective research areas for future strain engineering studies in 2DNMs.

Nanoindentation analysis of oriented polypropylene: Influence of elastic properties in tension and compression

Polypropylene has been oriented by solid-phase deformation processing to draw ratios up to ∼16, increasing tensile stiffness along the draw direction by factors up to 12. Nanoindentation of these materials showed that moduli obtained for indenter tip motion along the drawing direction (3) into to 1–2 plane (axial indentation) were up to 60% higher than for indenter tip motion along the 2 direction into the 1–3 plane (transverse indentation). In static tests, tensile and compressive determinations of elastic modulus gave results differing by factors up to ∼5 for strain along the draw direction. A material model incorporating both orthotropic elasticity and tension/compression asymmetry was developed for use with Finite Element simulations. Elastic constants for the oriented polypropylene were obtained by combining static testing and published ultrasonic data, and used as input for nanoindentation simulations that were quantitatively successful. The significance of the tension/compression asymmetry was demonstrated by comparing these predictions with those obtained using tensile data only, which gave predictions of indentation modulus higher by up to 70%.

Influence of Tcnq Loading on the Mechanical Properties of Metal-Organic Framework Films

Bulk Metal-organic-framework (MOF) materials are designed scaffold-like compounds that consist of metal ions connected by organic ligands, forming highly ordered porous structures. Bulk MOFs yield typically powders with micron sized crystallites, which limit their applications. Many practical applications require instead suitably uniform MOF films. In recent years, efforts to grow highly oriented Surface Anchored Metal-Organic Framework (SURMOF) films of Cu3(BTC)2 (BTC: benzenetricarboxylic acid), known as HKUST-1 MOF, have been successful by utilizing a liquid phase epitaxy (LPE) spray method on surface functionalized gold coated silicon substrates. Originally MOFs were intended for gas storage due to high storage capacity inside the porous MOF bulk material. New potential device applications in thermoelectrics and microelectronics [1,2] were explored based on the premise that the electrical properties of the otherwise insulating MOF can be custom tailored by loading the porous MOF structure with metal clusters. For modulating the electrical conductivity of MOF films the most effective method is to infiltrate guest molecules inside the porous framework to transport ions, and hence to modulate the electrical properties of the host framework. Early work focused on the mechanical properties of pristine SURMOF films [3,4]. In this study, we have investigated the influence of loading pristine highly oriented MOF films with TCNQ on the mechanical properties by utilizing nanoindentation methods to measure the hardness and modulus of SURMOF films with and without TCNQ loading. Our findings reveal that undoped pristine SURMOF samples are relatively harder with an intrinsic hardness of ~ 1.6 GPa in the upper third of the MOF film. The hardness of the pristine SURMOF samples is steadily increasing as a function of the penetration depth. The hardness is approaching the substrate Si hardness as the indenter tip approaches the substrate. The measured modulus values of the pristine SURMOF samples are also steadily increasing as a function of the indentation depth. However, the TCNQ loaded SURMOF samples differ markedly and do not show a substrate effect. Interestingly the loaded SURMOF films are much softer and exhibit a constant hardness as a function of depth of ~ 0.5 GPa. As the indenter approaches the Si substrate there is no measurable change in the constant hardness and an absence of a substrate effect. The modulus values of TCNQ loaded SURMOF films is also significantly lower compared with the pristine sample and remains constant as a function of indenter penetration depth. Thus, loading the porous scaffolding of SURMOF films with TCNQ did not only alter the electrical conductivity but triggers marked changes in the mechanical properties, i.e., softening effect. This softening effect in TCNQ infiltrated MOF films can be accounted for by a widening of the lattice constant und the concomitant introduction of lattice defects that weaken the crystal and lead to a much lower hardness.

Measuring bone stiffness using spherical indentation.

ObjectivesBone material properties are a major determinant of bone health in older age, both in terms of fracture risk and implant fixation, in orthopaedics and dentistry. Bone is an anisotropic and hierarchical material so its measured material properties depend upon the scale of metric used. The scale used should reflect the clinical problem, whether it is fracture risk, a whole bone problem, or implant stability, at the millimetre-scale. Indentation, an engineering technique involving pressing a hard-tipped material into another material with a known force, may be able to assess bone stiffness at the millimetre-scale (the apparent elastic modulus). We aimed to investigate whether spherical-tip indentation could reliably measure the apparent elastic modulus of human cortical bone.Materials and methodsCortical bone samples were retrieved from the femoral necks of nineteen patients undergoing total hip replacement surgery (10 females, 9 males, mean age: 69 years). The samples underwent indentation using a 1.5 mm diameter, ruby, spherical indenter tip, with sixty indentations per patient sample, across six locations on the bone surfaces, with ten repeated indentations at each of the six locations. The samples then underwent mechanical compression testing. The repeatability of indentation measurements of elastic modulus was assessed using the co-efficient of repeatability and the correlation between the bone elastic modulus measured by indentation and compression testing was analysed by least-squares regression.ResultsIn total, 1140 indentations in total were performed. Indentation was found to be repeatable for indentations performed at the same locations on the bone samples with a mean co-efficient of repeatability of 0.4 GigaPascals (GPa), confidence interval (C.I): 0.33–0.42 GPa. There was variation in the indentation modulus results between different locations on the bone samples (mean co-efficient of repeatability: 3.1 GPa, C.I: 2.2–3.90 GPa). No clear correlation was observed between indentation and compression values of bone elastic modulus (r = 0.33, p = 0.17). The mean apparent elastic modulus obtained by spherical indentation was 9.9 GPa, the standard deviation for each indent cycle was 0.11 GPa, and the standard deviation between locations on the same sample was 1.01 GPa. The mean compression apparent elastic modulus was 4.42 GPa, standard deviation 1.02 GPa.DiscussionSpherical-tip indentation was found to be a repeatable test for measuring the elastic modulus of human cortical bone, demonstrated by a low co-efficient of repeatability in this study. It could not, however, reliably predict cortical bone elastic modulus determined by platens compression testing in this study. This may be due to indentation only probing mechanical properties at the micro-scale while platens compression testing assesses millimetre length-scale properties. Improvements to the testing technique, including the use of a larger diameter spherical indenter tip, may improve the measurement of bone stiffness at the millimetre scale and should be investigated further.

Enhancing adhesion strength of a-C:H:Cu composite coatings on Ti6Al4V by graded copper deposition in a rf-PVD/PECVD hybrid process

In the medical field, poor coating adhesion often results in the limitation of the application of coating systems on implant devices. This limitation indicates a need for the development of improved techniques and methods to overcome this issue. In this paper, the use of graded copper deposition to enhance the adhesion strength of a-C:H:Cu composite coatings for biomedical applications grown by a rf-PVD/PECVD hybrid process is examined. In situ OES measurements give deeper insight in the amount of Cu(I) ions in the process plasma during deposition process which is used to control XCu in the a-C:H:Cu coatings cross section. With this knowledge, the critical load of a-C:H:Cu coatings can be enhanced from 50 mN to >300 mN depending on overall XCu. The influence of substrate bias and Cu mole fraction (XCu = (4 … 84) %) on the mechanical and structural properties of hydrogenated amorphous carbon films (a-C:H) is investigated. Polished Ti6Al4V and Si were used as substrate materials. The coating's microstructure was determined by STEM imaging and X-ray diffraction, while mechanical and adhesion properties were examined by nanoindentation and nano scratch tests using a Berkovich indenter tip. The nanocomposite coatings consist of Cu nanoparticles embedded in an a-C:H matrix. Their morphology changes to a columnar structure with Cu nanorods for high XCu when applying a substrate bias voltage of −50 V. For Cu free a-C:H coatings, a maximum hardness of 13.6 GPa was measured which is lowered down to 4 GPa for a-C:H:Cu coatings with XCu = 84%. A significant strengthening of the coatings is observed when applying a subsequent substrate bias voltage of −50 V, which further enhances scratch resistance.

Influencing the adhesion properties and wettability of mucin protein films by variation of the environmental pH

Mucins, the main component of the mucus secretions of goblet and epithelial cells, are known for exhibiting a different behaviour in accordance with their surrounding environment (i.e. among others the environmental pH), which induces a drastic change in their measured mechanical properties. In this work, we have first employed Atomic Force Microscopy (AFM) in Force Spectroscopy mode to evaluate the adhesion of porcine mucin films at the nanoscale, and the changes caused in this particular factor by a pH variation between 7.0 and 4.0, both quite common values in biological conditions. Measurements also involved additional varying factors such as the indenting tip chemistry (hydrophobic vs hydrophilic), its residence time on the measured film (0, 1 and/or 2 seconds), and increasing pulling rates (ranging from 0.1 up to 10 µm/s). A second approach regarded the macroscale behaviour of the films, due to their potential applicability in the development of a new set of stimuli-responsive biomaterials. This was possible by means of complementary Wilhelmy plate method (to test the wetting properties) and cell proliferation studies on films previously exposed to the corresponding pH solution. According to our results, treatment with lowest pH (4.0) provides porcine mucin with a more hydrophilic character, showing a much stronger adhesion for analogous chemistries, as well as enhanced capability for cell attachment and proliferation, which opens new pathways for their future use and consideration as scaffold-forming material.

Asymmetric indentation of an elastic beam by a rigid cylinder

A simple method is proposed to analyze asymmetric indentation of a supported elastic beam by a rigid cylinder placed at an arbitrary position on the beam. A key advantage of our method over existing methods is that a single governing Kerr-type differential equation, which gives the relation between contact pressure and the normal deflection of the pressured surface of the beam, is applied both inside and outside the contact zone. With the present method, the two ends of contact zone will be determined as part of the solution, in contrast to the existing related studies which were all based on a simplifying assumption that the contact zone is geometrically symmetric about the tip of the indenter. Indeed, our results confirm that the contact zone is generally geometrically non-symmetric about the indenter tip, and even the whole contact zone can locate on one side of the indenter tip in some cases. Asymmetric indentation behaviors are demonstrated with numerical examples, with an emphasis on asymmetric contact pressure distribution inside the contact zone. Explicit formulas for the contact-zone width–displacement, force–displacement and moment–displacement relations are derived and illustrated with numerical examples. In particular, the present model predicts that as the indentation displacement exceeds a critical value, the indenter will lose contact with the beam inside the contact zone, which implies that the actual contact zone becomes two separate contact strips. Validity and accuracy of the present method are demonstrated by comparing its predictions with known results for some special cases. New features of asymmetric indentation as compared to symmetric indentation are summarized at the end of the paper.