Nanoindentation, also called intrumented indentation testing, is a variety of indentation hardness tests applied to small volumes. Indentation is perhaps the most commonly applied means of testing the mechanical properties of materials. The nanoindentation technique was developed in the mid-1970s to measure the hardness of small volumes of material.
In a traditional indentation test (macro or micro indentation), a hard tip whose mechanical properties are known (frequently made of a very hard material like diamond) is pressed into a sample whose properties are unknown. The load placed on the indenter tip is increased as the tip penetrates further into the specimen and soon reaches a user-defined value. At this point, the load may be held constant for a period or removed. The area of the residual indentation in the sample is measured and the hardness, , is defined as the maximum load, , divided by the residual indentation area, :
For most techniques, the projected area may be measured directly using light microscopy. As can be seen from this equation, a given load will make a smaller indent in a "hard" material than a "soft" one.
This technique is limited due to large and varied tip shapes, with indenter rigs which do not have very good spatial resolution (the location of the area to be indented is very hard to specify accurately). Comparison across experiments, typically done in different laboratories, is difficult and often meaningless. Nanoindentation improves on these macro- and micro-indentation tests by indenting on the nanoscale with a very precise tip shape, high spatial resolutions to place the indents, and by providing real-time load-displacement (into the surface) data while the indentation is in progress.
In nanoindentation small loads and tip sizes are used, so the indentation area may only be a few square micrometres or even nanometres. This presents problems in determining the hardness, as the contact area is not easily found. Atomic force microscopy or scanning electron microscopy techniques may be utilized to image the indentation, but can be quite cumbersome. Instead, an indenter with a geometry known to high precision (usually a Berkovich tip, which has a three-sided pyramid geometry) is employed. During the course of the instrumented indentation process, a record of the depth of penetration is made, and then the area of the indent is determined using the known geometry of the indentation tip. While indenting, various parameters such as load and depth of penetration can be measured. A record of these values can be plotted on a graph to create a load-displacement curve (such as the one shown in Figure 1). These curves can be used to extract mechanical properties of the material.
The slope of the curve, , upon unloading is indicative of the stiffness of the contact. This value generally includes a contribution from both the material being tested and the response of the test device itself. The stiffness of the contact can be used to calculate the reduced Young's modulus :
Where is the projected area of the indentation at the contact depth , and is a geometrical constant on the order of unity. is often approximated by a fitting polynomial as shown below for a Berkovich tip:
Where for a Berkovich tip is 24.5 while for a cube corner (90°) tip is 2.598. The reduced modulus is related to Young's modulus of the test specimen through the following relationship from contact mechanics:
Here, the subscript indicates a property of the indenter material and is Poisson's ratio. For a diamond indenter tip, is 1140 GPa and is 0.07. Poisson’s ratio of the specimen, , generally varies between 0 and 0.5 for most materials (though it can be negative) and is typically around 0.3.
There are two different types of hardness that can be obtained from a nano indenter: one is as in traditional macroindentation tests where one attains a single hardness value per experiment; the other is based on the hardness as the material is being indented resulting in hardness as a function of depth.
The hardness is given by the equation above, relating the maximum load to the indentation area. The area can be measured after the indentation by in-situ atomic force microscopy, or by 'after-the event' optical (or electron) microscopy. An example indentation image, from which the area may be determined, is shown at right.
Some nanoindenters use an area function based on the geometry of the tip, compensating for elastic load during the test. Use of this area function provides a method of gaining real-time nanohardness values from a load-displacement graph. However, there is some controversy over the use of area functions to estimate the residual areas versus direct measurement.[ citation needed ] An area function typically describes the projected area of an indent as a 2nd-order polynomial function of the indenter depth . When too many coefficients are used, the function will begin to fit to the noise in the data, and inflection points will develop. If the curve can fit well with only two coefficients, this is the best. However, if many data points are used, sometimes all 6 coefficients will need to be used to get a good area function. Typically, 3 or 4 coefficients works well. Exclusive application of an area function in the absence of adequate knowledge of material response can lead to misinterpretation of resulting data. Cross-checking of areas microscopically is to be encouraged.
The strain-rate sensitivity of the flow stress is defined as
where is the flow stress and is the strain rate produced under the indenter. For nanoindentation experiments which include a holding period at constant load (i.e. the flat, top area of the load-displacement curve), can be determined from
The subscripts indicate these values are to be determined from the plastic components only.
Interpreted loosely as the volume swept out by dislocations during thermal activation, the activation volume is
where is the temperature and kB is Boltzmann's constant. From the definition of , it is easy to see that .
The construction of a depth-sensing indentation system is made possible by the inclusion of very sensitive displacement and load sensing systems. Load transducers must be capable of measuring forces in the micronewton range and displacement sensors are very frequently capable of sub-nanometer resolution. Environmental isolation is crucial to the operation of the instrument. Vibrations transmitted to the device, fluctuations in atmospheric temperature and pressure, and thermal fluctuations of the components during the course of an experiment can cause significant errors.
Dynamic nanoindentation or continuous stiffness measurement (CSM, also offered commercially as CMX, dynamics...), introduced in 1989, is a significant improvement over the quasi-static mode described above. It consists into overlapping a very small, fast (> 40 Hz) oscillation onto the main loading signal and evaluate the magnitude of the resulting partial unloadings by a lock-in amplifier, so as to quasi-continuously determine the contact stiffness. This allows for the continuous evaluation of the hardness and Young's modulus of the material over the depth of the indentation, which is of great advantage with coatings and graded materials. The CSM method is also pivotal for the experimental determination of the local creep and strain-rate dependent mechanical properties of materials, as well as the local damping of visco-elastic materials. The harmonic amplitude of the oscillations is usually chosen around 2 nm (RMS), which is a trade-off value avoiding an underestimation of the stiffness due to the "dynamic unloading error" or the "plasticity error" during measurements on materials with unusually high elastic-to-plastic ratio (E/H > 150), such as soft metals.
The ability to conduct nanoindentation studies with nanometer depth, and sub-nanonewton force resolution is also possible using a standard AFM setup. The AFM allows for nanomechanical studies to be conducted alongside topographic analyses, without the use of dedicated instruments. Load-displacement curves can be collected similarly for a variety of materials - provided that they are softer than the AFM tip - and mechanical properties can be directly calculated from these curves.. Conversely, some commercial nanoindentation systems offer the possibility to use a piezo-driven stage to image the topography of residual indents with the nanoindenter tip.
The indentation curves have often at least thousands of data points. The hardness and elastic modulus can quickly be calculated by using a programming language or a spreadsheet. Instrumented indentation testing machines come with the software specifically designed to analyze the indentation data from their own machine. The Indentation Grapher (Dureza) software is able to import text data from several commercial machines or custom made equipment. displacement so that the data passes through the origin. Then select the power law equation from the graphing options.Spreadsheet programs such as MS-Excel or OpenOffice Calculate do not have the ability to fit to the non-linear power law equation from indentation data. A linear fit can be done by offset
The Martens hardness, , is a simple software for any programmer having minimal background to develop. The software starts by searching for the maximum displacement, , point and maximum load, .
The displacement is used to calculate the contact surface area, , based on the indenter geometry. For a perfect Berkovich indenter the relationship is .
The indentation hardness, is defined slightly different.
Here, the hardness is related to the projected contact area .
As the indent size decreases the error caused by tip rounding increases. The tip wear can be accounted for within the software by using a simple polynomial function. As the indenter tip wears the value will increase. The user enters the values for and based on direct measurements such as SEM or AFM images of the indenter tip or indirectly by using a material of known elastic modulus or an atomic force microscope (AFM) image of an indentation.
Calculating the elastic modulus with software involves using software filtering techniques to separate the critical unloading data from the rest of the load-displacement data. The start and end points are usually found by using user defined percentages. This user input increases the variability because of possible human error. It would be best if the entire calculation process was automatically done for more consistent results. A good nanoindentation machine prints out the load unload curve data with labels to each of the segments such as loading, top hold, unload, bottom hold, and reloading. If multiple cycles are used then each one should be labeled. However mores nanoindenters only give the raw data for the load-unload curves. An automatic software technique finds the sharp change from the top hold time to the beginning of the unloading. This can be found by doing a linear fit to the top hold time data. The unload data starts when the load is 1.5 times standard deviation less than the hold time load. The minimum data point is the end of the unloading data. The computer calculates the elastic modulus with this data according to the Oliver—Pharr (nonlinear). The Doerner-Nix method is less complicated to program because it is a linear curve fit of the selected minimum to maximum data. However, it is limited because the calculated elastic modulus will decrease as more data points are used along the unloading curve. The Oliver-Pharr nonlinear curve fit method to the unloading curve data where is the depth variable, is the final depth and and are constants and coefficients. The software must use a nonlinear convergence method to solve for , and that best fits the unloading data. The slope is calculated by differentiating at the maximum displacement.
An image of the indent can also be measured using software. The atomic force microscope (AFM) scans the indent. First the lowest point of the indentation is found. Make an array of lines around the using linear lines from indent center along the indent surface. Where the section line is more than several standard deviations (>3 ) from the surface noise the outline point is created. Then connect all of the outline points to build the entire indent outline. This outline will automatically include the pile-up contact area.
For nanoindentation experiments performed with a conical indenter on a thin film deposited on a substrate or on a multilayer sample, the NIMS Matlab toolboxis useful for load-displacement curves analysis and calculations of Young's modulus and hardness of the coating . In the case of pop-in, the PopIn Matlab toolbox is a solution to analyze statistically pop-in distribution and to extract critical load or critical indentation depth, just before pop-in . Finally, for indentation maps obtained following the grid indentation technique, the TriDiMap Matlab toolbox offers the possibility to plot 2D or 3D maps and to analyze statistically mechanical properties distribution of each constituent, in case of a heterogeneous material by doing deconvolution of probability density function .
Molecular dynamics (MD) has been a very powerful technique to investigate the nanoindentation at atomic scale. For instance, Alexey et alemployed MD to simulate the nanoindentation process of a titanium crystal, dependence of deformation of the crystalline structure on the type of the indenter is observed, which is very hard to harvest in experiment. Tao et al performed MD simulations of nanoindentation on Cu/Ni nanotwinned multilayers films using a spherical indenter and investigated the effects of hetero-twin interface and twin thickness on hardness. Recently, a review paper by Carlos et al is published upon the atomistic studies of nanoindentation. This review covers different nanoindentation mechanisms and effects of surface orientation, crystallography (fcc, bcc, hcp, etc), surface and bulk damage on plasticity. All of the MD-obtained results are very difficult to be achieved in experiment due to the resolution limitation of structural characterization techniques. Among various MD simulation software, such as GROMACS, Xenoview, Amber, etc., LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator), which is developed by Sandia National Laboratories, is the most widely used for simulation. An interaction potential and an input file including information of atom ID, coordinates, charges, ensemble, time step, etc are fed to the simulator, and then running could be executed. After specified running timesteps, information such as energy, atomic trajectories, and structural information (such as coordination number) could be output for further analysis, which makes it possible to investigate the nanoindentation mechanism at atomic-scale. Another interesting Matlab toolbox called STABiX has been developed to quantify slip transmission at grain boundaries by analyzing indentation experiments in bicrystal .
Nanoindentation is a robust technique for determination of mechanical properties. By combining the application of low loads, measuring the resulting displacement, and determining the contact area between the tip of the indenter and the sample a wide range of mechanical properties are able to be measured.The application that drove the innovation of the technique is testing thin film properties for which conventional testing are not feasible. Conventional mechanical testing such as tensile testing or dynamic mechanical analysis (DMA) can only return the average property without any indication of variability across the sample. However, nanoindentation can be used for determination of local properties of homogeneous as well as heterogeneous materials. The reduction in sample size requirements has allowed the technique to become broadly applied to products where the manufactured state does not present enough material for microhardness testing. Applications in this area include medical implants, consumer goods, and packaging. Alternative uses of the technique are used to test MEMs devices by utilizing the low-loads and small scale displacements the nanoindenter is capable of.
Conventional nanoindentation methods for calculation of Modulus of elasticity (based on the unloading curve) are limited to linear, isotropic materials.
Problems associated with the "pile-up" or "sink-in" of the material on the edges of the indent during the indentation process remain a problem that is still under investigation. It is possible to measure the pile-up contact area using computerized image analysis of atomic force microscope (AFM) images of the indentations.This process also depends on the linear isotropic elastic recovery for the indent reconstruction.
Nanoindentation of soft material has intrinsic challenges due to adhesion, surface detection and tip dependency of results. There is an ongoing research to overcome such problems.
Two critical issues need to be considered when attempting nanoindentation measurements on soft materials: stiffness and viscoelasticity.
The first is the requirement that in any force-displacement measurement platform the stiffness of the machine () must approximately match the stiffness of the sample (), at least in order of magnitude. If is too high, then the indenter probe will simply run through the sample without being able to measure the force. On the other hand, if is too low, then the probe simply will not indent into the sample, and no reading of the probe displacement can be made. For samples that are very soft, the first of these two possibilities is likely.
The stiffness of a sample is given by
where is the size of the contact region between the indenter and the sample, and is the sample’s elastic modulus. Typical atomic-force microscopy (AFM) cantilevers have in the range 0.05 to 50 N/m, and probe size in the range ~10 nm to 1 μm. Commercial nanoindenters are also similar. Therefore, if ≈, then a typical AFM cantilever-tip or a commercial nanoindenter can only measure in the ~kPa to GPa range. This range is wide enough to cover most synthetic materials including polymers, metals and ceramics, as well as a large variety of biological materials including tissues and adherent cells. However, there may be softer materials with moduli in the Pa range, such as floating cells, and these cannot be measured by an AFM or a commercial nanoindenter.
To measure in the Pa range, “pico-indentation” using an optical tweezers system is suitable. Here, a laser beam is used to trap a translucent bead which is then brought into contact with the soft sample so as to indent it . The trap stiffness () depends on the laser power and bead material, and a typical value is ~50 pN/μm. The probe size can be a micron or so. Then the optical trap can measure (≈/)in the Pa range.
The second issue concerning soft samples is their viscoelasticity. Methods to handle viscoelasticity include the following.
In the classical treatment of viscoelasticity, the load-displacement (P-h) response measured from the sample is fitted to predictions from an assumed constitutive model (e.g. the Maxwell model) of the material comprising spring and dashpot elements. Such an approach can be very time consuming, and cannot in general prove the assumed constitutive law in an unambiguous manner.
Dynamic indentation with an oscillatory load can be performed, and the viscoelastic behavior of the sample is presented in terms of the resultant storage and loss moduli, often as variations over the load frequency. However, the storage and loss moduli obtained this way are not intrinsic material constants, but depend on the oscillation frequency and the indenter probe geometry.
A rate-jump method can be used to return an intrinsic elastic modulus of the sample that is independent of the test conditions ij=ij(kl) but stress kl is continuous across the step change ∆ij in the stress rate field kl at tc, there will not be any corresponding change in the strain rate field ij across the dashpots. However, because the linear elastic springs are described by relations of the form ij=Sikjlkl where Sikjl are elastic compliances, a step change ∆ij across the springs will result according to. In this method, a constitutive law comprising any network of (in general) non-linear dashpots and linear elastic springs is assumed to hold within a very short time window about the time instant tc at which a sudden step change in the loading rate is applied on the sample. Since the dashpots are described by relations of the form
The last equation indicates that the fields ∆kl and ∆ij can be solved as a linear elastic problem with the elastic spring elements in the original viscoelastic network model while the dashpot elements are ignored. The solution for a given test geometry is a linear relation between the step changes in the load and displacement rates at tc, and the linking proportionality constant is a lumped value of the elastic constants in the original viscoelastic model. Fitting such a relation to experimental results allows this lumped value to be measured as an intrinsic elastic modulus of the material.
Specific equations from this rate-jump method have been developed for specific test platforms. For example, in depth-sensing nanoindentation, the elastic modulus and hardness are evaluated at the onset of an unloading stage following a load-hold stage. Such an onset point for unloading is a rate-jump point, and solving the equation ij=Sikjlkl across this leads to the Tang-Ngan method of viscoelastic correction
where S = dP/dh is the apparent tip-sample contact stiffness at the onset of unload, is the displacement rate just before the unload, is the unloading rate, and is the true (i.e. viscosity-corrected) tip-sample contact stiffness which is related to the reduced modulus and the tip-sample contact size by the Sneddon relation . The contact size a can be estimated from a pre-calibrated shape function = of the tip, where the contact depth is obtainable using the Oliver—Pharr relation with the apparent contact stiffness replaced by the true stiffness :
where is a factor depending on the tip (say, 0.72 for Berkovich tip).
While nanoindentation testing can be relatively simple, the interpretation of results is challenging. One of the main challenges is the use of proper tip depending on the application and proper interpretation of the results. For instance, it has been shown that the elastic modulus can be tip dependent.
Young's modulus, or the Young modulus, is a mechanical property that measures the stiffness of a solid material. It defines the relationship between stress and strain in a material in the linear elasticity regime of a uniaxial deformation.
The Brinell scale characterizes the indentation hardness of materials through the scale of penetration of an indenter, loaded on a material test-piece. It is one of several definitions of hardness in materials science.
Stiffness is the extent to which an object resists deformation in response to an applied force.
Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.
The Vickers hardness test was developed in 1921 by Robert L. Smith and George E. Sandland at Vickers Ltd as an alternative to the Brinell method to measure the hardness of materials. The Vickers test is often easier to use than other hardness tests since the required calculations are independent of the size of the indenter, and the indenter can be used for all materials irrespective of hardness. The basic principle, as with all common measures of hardness, is to observe a material's ability to resist plastic deformation from a standard source. The Vickers test can be used for all metals and has one of the widest scales among hardness tests. The unit of hardness given by the test is known as the Vickers Pyramid Number (HV) or Diamond Pyramid Hardness (DPH). The hardness number can be converted into units of pascals, but should not be confused with pressure, which uses the same units. The hardness number is determined by the load over the surface area of the indentation and not the area normal to the force, and is therefore not pressure.
Work hardening, also known as strain hardening, is the strengthening of a metal or polymer by plastic deformation. Work hardening may be desirable, undesirable, or inconsequential, depending on the context.
Nanotribology is the branch of tribology that studies friction, wear, adhesion and lubrication phenomena at the nanoscale, where atomic interactions and quantum effects are not negligible. The aim of this discipline is characterizing and modifying surfaces for both scientific and technological purposes.
Hardness is a measure of the resistance to localized plastic deformation induced by either mechanical indentation or abrasion. Some materials are harder than others. Macroscopic hardness is generally characterized by strong intermolecular bonds, but the behavior of solid materials under force is complex; therefore, there are different measurements of hardness: scratch hardness, indentation hardness, and rebound hardness.
Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, stretched or generally deformed in any manner. Elasticity theory primarily develops formalisms for the mechanics of solid bodies and materials. The elastic potential energy equation is used in calculations of positions of mechanical equilibrium. The energy is potential as it will be converted into other forms of energy, such as kinetic energy and sound energy, when the object is allowed to return to its original shape (reformation) by its elasticity.
The three-point bending flexural test provides values for the modulus of elasticity in bending , flexural stress , flexural strain and the flexural stress–strain response of the material. The main advantage of a three-point flexural test is the ease of the specimen preparation and testing. However, this method has also some disadvantages: the results of the testing method are sensitive to specimen and loading geometry and strain rate.
The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material. The theoretical concept of J-integral was developed in 1967 by Cherepanov and in 1968 by Jim Rice independently, who showed that an energetic contour path integral was independent of the path around a crack.
The Shore durometer is a device for measuring the hardness of a material, typically of polymers, elastomers, and rubbers.
Ceramography is the art and science of preparation, examination and evaluation of ceramic microstructures. Ceramography can be thought of as the metallography of ceramics. The microstructure is the structure level of approximately 0.1 to 100 µm, between the minimum wavelength of visible light and the resolution limit of the naked eye. The microstructure includes most grains, secondary phases, grain boundaries, pores, micro-cracks and hardness microindentions. Most bulk mechanical, optical, thermal, electrical and magnetic properties are significantly affected by the microstructure. The fabrication method and process conditions are generally indicated by the microstructure. The root cause of many ceramic failures is evident in the microstructure. Ceramography is part of the broader field of materialography, which includes all the microscopic techniques of material analysis, such as metallography, petrography and plastography. Ceramography is usually reserved for high-performance ceramics for industrial applications, such as 85–99.9% alumina (Al2O3) in Fig. 1, zirconia (ZrO2), silicon carbide (SiC), silicon nitride (Si3N4), and ceramic-matrix composites. It is seldom used on whiteware ceramics such as sanitaryware, wall tiles and dishware.
The Meyer hardness test is a hardness test based upon projected area of an impression. The hardness, , is defined as the maximum load, divided by the projected area of the indent, .
A nanoindenter is the main component for indentation hardness tests used in nanoindentation. Since the mid-1970s nanoindentation has become the primary method for measuring and testing very small volumes of mechanical properties. Nanoindentation, also called depth sensing indentation or instrumented indentation, gained popularity with the development of machines that could record small load and displacement with high accuracy and precision. The load displacement data can be used to determine modulus of elasticity, hardness, yield strength, fracture toughness, scratch hardness and wear properties.
In materials science, a general rule of mixtures is a weighted mean used to predict various properties of a composite material made up of continuous and unidirectional fibers. It provides a theoretical upper- and lower-bound on properties such as the elastic modulus, mass density, ultimate tensile strength, thermal conductivity, and electrical conductivity. In general there are two models, one for axial loading, and one for transverse loading.
Materials that are used for biomedical or clinical applications are known as biomaterials. The following article deals with fifth generation biomaterials that are used for bone structure replacement. For any material to be classified for biomedical application three requirements must be met. The first requirement is that the material must be biocompatible; it means that the organism should not treat it as a foreign object. Secondly, the material should be biodegradable ; the material should harmlessly degrade or dissolve in the body of the organism to allow it to resume natural functioning. Thirdly, the material should be mechanically sound; for the replacement of load bearing structures, the material should possess equivalent or greater mechanical stability to ensure high reliability of the graft.
Plasticity theory for rocks is concerned with the response of rocks to loads beyond the elastic limit. Historically, conventional wisdom has it that rock is brittle and fails by fracture while plasticity is identified with ductile materials. In field scale rock masses, structural discontinuities exist in the rock indicating that failure has taken place. Since the rock has not fallen apart, contrary to expectation of brittle behavior, clearly elasticity theory is not the last work.
Flow plasticity is a solid mechanics theory that is used to describe the plastic behavior of materials. Flow plasticity theories are characterized by the assumption that a flow rule exists that can be used to determine the amount of plastic deformation in the material.
The Korsunsky work-of-indentation approach is a method of extracting a value of hardness for a small volume of material from indentation test data. Instead of relying on measurements or assumptions pertaining to the area of contact between indenter and sample, the method uses the load-displacement data registered in the continuously recorded indentation testing (CRIT) that is particularly widely applied in nanoindentation experiments. In particular, the method re-defines hardness and expresses it in terms of the energy (work) associated with indenting the surface of a material by the probe. The work-of-indentation used in the analysis may refer to the total, elastic or dissipated energy, depending on the formulation. The approach was found to be particularly useful in the analysis of thin coatings, nano-multi-layers, nanoscale features.
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