Experimental Mechanics DOI 10.1007/s11340-010-9373-z Determination of Mechanical Properties of Sand Grains by Nanoindentation N.P. Daphalapurkar & F. Wang & B. Fu & H. Lu & R. Komanduri Received: 3 August 2009 / Accepted: 24 May 2010 # Society for Experimental Mechanics 2010 Abstract Determination of the mechanical properties of individual sand grains by conventional material testing methods at the macroscale is somewhat difficult due to the sizes of the individual sand particles (a few μm to mm). In this paper, we used the nanoindentation technique with a Berkovich tip to measure the Young’s modulus, hardness, and fracture toughness. An inverse problem solving approach was adopted to determine the stress-strain relationship of sand at the granular level using the finite element method. A cube-corner indenter tip was used to generate radial cracks, the lengths of which were used to determine the fracture toughness. Scatter in the data was observed, as is common with most brittle materials. In order to consider the overall mechanical behavior of the sand grains, statistical analysis of the mechanical properties data (including the variability in the properties) was conducted N.P. Daphalapurkar Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA e-mail:
[email protected] F. Wang : H. Lu (SEM member) Department of Mechanical Engineering, The University of Texas at Dallas, Richardson, TX 75080, USA H. Lu e-mail:
[email protected] B. Fu : R. Komanduri (*) School of Mechanical and Aerospace Engineering, Oklahoma State University, 218 Engineering North, Stillwater, OK 74078, USA e-mail:
[email protected] B. Fu e-mail:
[email protected] using the Weibull distribution function. This data can be used in the mesoscale simulations. Keywords Sand . Mechanical properties . Nanoindentation . Young’s modulus . Hardness . Stress-strain . Fracture toughness . Particulate mechanics Introduction Granular materials, such as sand, are conglomerates of discrete particles held together (but not bonded) with significant void space (35–65%). They are unique in that they behave in some respects similar to the other familiar forms of matter, namely, solids, liquids, and gases and in other respects in a dissimilar form. For example, they pack like solids but flow like liquids. Like liquids, they take the shape of the container but unlike them they can adopt to a variety of shapes when they are free standing. Similarly, like gases, they are made of discrete particles with negligible cohesive forces between them. Like solid, they can support load, but unlike a solid, they hardly support any tensile load. In view of their unique behavior, some consider granular materials as the fifth state of matter, alongside, solids, liquids, gases, and plasma. They cannot strictly be modeled as a continuum, yet it is done by considering the movement of particles in the void space under load, akin to the deformation and flow in solids. Sand is formed largely by erosion and disintegration of larger rocks into particles by natural forces, such as wind, pressure, water, ice, friction, and heat. Over millions of years, such processes have led to the formation of sand of various grain sizes ranging from a fraction of a micrometer to several millimeters. Investigation of the mechanical behavior of sand from granular (mesoscale) to macro it becomes possible to characterize such materials and extract the moduli and hardness values at different depths of indentation. ridges. and the size (typically a few mm) of the influential shearing zones in tectonic plates in sliding during earthquakes and landslides. Mechanical properties. are of particular importance since their individual behavior dictates the overall behavior of sand at the macroscale. size. in sandbags to prevent soil erosion at the banks and act as temporary dams against floods. Moreover. Such an ease in the characterization of length-scale dependent material properties using nanoindentation is in contrast with the tedious procedure involved in using conventional tensile testing on small samples. fracture toughness. Furthermore. and shear band formation by Wang and Lade [5]. by using the experimental load-depth information in combination with Finite Element (FE) numerical simulations. hardness. As a result. especially in the soil mechanics area. However. However. Additionally. measurement of mechanical properties can be carried out using tensile/ compression testing machine. wires. Sand is often used to provide ballistic protection for military and national security applications. such as thin films. and microcracks which could adversely effect the yield properties. simple shear tests were conducted by Haythornthwaite [6] to determine shear locus of sand. This is partially due to the emergence of new technological instruments.e. the localization of deformation [10]. difficulties arise in the preparation of the sample and/or holding smaller grains in the material testing machine prior to subjecting them to compression or pulled in tension for the measurements of mechanical properties.g. As far as larger (on the order of a centimeter or more) sand particles (rocks) are concerned. viscoelastic). and stress-strain relations of sand grains. by adopting an inverse methodology approach. and as a construction material in civil engineering structures. we used the nanoindentation technique to determine the force-displacement relation and extracted the mechanical properties. such as nanoindentation. Consequently. atomic force microscopy (AFM). and fracture of granular materials under load and ultimately realize the underlying science-base for large scale phenomena. mechanical properties of larger grains cannot be directly used to imply the behavior of finer grains. i. such as the Young’s modulus. wherein the materials exhibit length-scale dependent behavior. [1] and Lade and Duncan [2] conducted triaxial compression tests on sand to determine the nature of deformation and attempted to describe it using a constitutive law. Considerable research had been reported in the literature on the mechanical behavior of sand. defect structure. flow. this field yet remains to be relatively insufficiently understood. the stresses are considered to propagate in a manner described by the wave-like (hyperbolic) equations [7–9] rather than by the elliptic equations of static elasticity. for smaller systems at the granular level. such as the motion of sand particles by an impacting solid.Exp Mech (continuum) scale can contribute towards a fundamental understanding of the underlying mechanisms of deformation. The instrumented nanoindentation is widely accepted as a standardized testing method for the characterization of mechanical properties for elastic-plastic materials. such as the formation of force-chains is more pronounced and deformation mechanisms at meso/micro scales need to be considered [11]. we predicted the stress-strain response. extracting the material properties by indentation at depths < 100 nm would essentially provide mechanical properties at nanoscale. The material properties extracted will be a function of the maximum indentation depth reached or the deformation zone beneath the indenter tip. the mechanical properties of the grains can vary with the mineral composition. By using the cube-corner of an indenter tip to induce cracks in the sand grain. However. we characterized the fracture behavior of sand particles. Triaxial compression tests were also carried out by Arthur and Menzies [3] to investigate anisotropy. X-ray microtomography as well as the availability of rapidly increasing computational power that has facilitated in addressing this complex problem. and coatings as well as for MEMS and NEMS components.. hardness. . softening and preshearing effects by Lade and Duncan [4]. Also. such as the Young’s modulus. has been an area of increasing interest especially in the past decade or so. nanoindentation has been used for the measurement of mechanical properties of different types of materials (e. Thus. Grain size plays an important role due to the fact that larger grains would contain higher number of defects in the form of voids. Hence. the mechanical behavior of granular or particulate materials. elastic-plastic. Brief Review of Literature It is well known that dense/highly compacted sand possess high compressive strength and high energy absorption capacity. The added advantage of such testing is that in some cases. such as sand. for small-scale systems. For the past two decades. and crystal orientation of sand. and stress-strain relationship of sand granules at microscale. nanoindentation technique is being widely used to extract mechanical properties of very small volume of materials. In this investigation. Poorooshasb et al. We conducted nanoindentations on several grains to assess the variance in the mechanical properties. The resolution in such a test can reach a fraction of a nanometer in displacement and µN in load. fracture toughness. and porosity under an optical microscope. In this investigation. available near the lakes in Stillwater. In addition to the measurement of the elastic modulus and hardness. The lengths of the cracks developed were used to predict the range of fracture toughness values for the sand grains. Statistical analysis was conducted to determine the variability in the mechanical properties of sand grains. However. Liao et al. This method is very well established and has been implemented in commercially instrumented nanoindenters for use on elastic-plastic mateFig. the elimination of the need for direct observation of the residual indentation impressions. 21]. OK was used in this investigation. an EDS . In all the cases. There have been many efforts on characterizing mechanical behavior. Additionally. nanoindentation has also been used to determine the fracture toughness of brittle materials [22. Ulm and Abousleiman [13] conducted nanoindentation measurements on shales. Nanoindentation Measurements A sample of sand. contrast. This procedure was found to be adequate for estimating the fracture toughness in homogeneous materials. Young’s modulus and hardness were obtained directly from the MTS TestWorks software output based on the analysis of unloading segment using the method developed by Oliver and Pharr [19]. and concluded that shales are nanogranular composite materials with their mechanical properties dictated by particle-to-particle contact and packing density. it is necessary to know the mechanical properties of individual grains. Instead. 19]. we conducted nanoindentation tests using a Berkovich indenter tip on several (∼250) individual sand grains of ∼1 mm size and recorded the loaddisplacement data. the unloading curve is used to extract the elastic properties of an elastic-plastic material.and nano. the data for larger sizes cannot be extended directly for individual sand grains. Figure 1(a) shows the SEM image of sand grains. The wide acceptance of nanoindentation technique stems from the improvements incorporated in the technique. A representative stress-strain curve was obtained from which the corresponding average modulus was obtained. The sand was washed and subsequently dried in an oven at ∼55°C. including. [12] conducted direct tensile tests on transversely isotropic cylindrical argillite specimens and reported the stress-strain relationships. They determined the material properties and found significant variations depending on the amount and orientation of the micro-fissures. namely. The mechanisms of deformation and fracture in the individual sand grains can be different from their larger counterparts. For example. using numerical techniques. elastic modulus and hardness by calculating the area of indent impression from the loading/unloading curves. Nanoindentation provides an effective technique for the measurement of local material response in terms of hardness and Young’s modulus at micro. the indent impressions were obtained using MTS NanoVision setup. in combination with experimental results have been developed effectively to measure the material properties and also the stress-strain relationships [20. a type of sedimentary rock. Nanoindents were also made using a cube-corner indenter to generate cracks for the estimation of fracture toughness.length scales (see for e.g. 1 (a) SEM image of sand grains. such as dynamic Young’s modulus and hardness have been well established [18. They exhibit different shapes. sizes. For example. It maybe noted that the sand sample used for nanoindentation has widely distributed sizes. Refs. 23]. This was used for solving the inverse problem to determine the stress-strain curve using FEM. mainly due to the difference in the length scales. For geomechanics applications and simulations. [14–17]). Inverse methodologies. Methods for measuring the elastic-plastic properties. failure behavior of rocks.Exp Mech Most of the literature on characterization of mechanical properties of sand or rocks has focused on either compressive behavior of sand as bulk or macroscale properties of rocks by pulling them in a tensile testing machine. (b) (color online) Magnified optical image of polished sand grains in an epoxy matrix rials. the unloading segment. After 10 min of shaking. The contact stiffness (S) is calculated from the slope of the initial unloading curve. Al. respectively. and is calculated based on the tip area function. Figure 2(a) shows an inverted image (3D) and a typical nanoindentation residual impression obtained using the NanoVision. The samples were polished using an alumina abrasive powder (from Buehler Inc. To obtain a smooth surface suitable for nanoindentation. An MTS Nano Indenter XP system was used for nanoindentation measurements. The load-displacement curves obtained thus are characteristic of that particular sand grain. and mineral content.Exp Mech spectrum was obtained in the SEM showing the presence of Si. It can be seen that no cracks were formed when indented with a Berkovich indenter-tip. the load was removed. nanoindentation tests were carried out on a polished.1 to 2 mm with an average grain size of 1 mm.) in a water slurry. were used in this investigation.5 mm) in three . is termed the loading segment. The sand grains were embedded in an epoxy matrix and mounted in a sample holder. Er Es Ei ð2Þ where Es and νs are the Young’s modulus and Poisson’s ratio of the specimen. The inverted image enables the determination of the depth of the indent more accurately and delineates its topographical features. the finest abrasive size used in the final polishing was 50 nm. defects such as small pits and ridges were observed. characterized by an increase in the load from zero to maximum. Analysis is carried out on the load-displacement output to determine the mechanical properties of the sand grains based on contact mechanics of nanoindentation. Once the user-defined maximum load was reached. relatively large sand grain (size ∼1. again in a controlled manner (linearly) and is termed. The initial part of the loaddisplacement curve. No particular analysis was carried out to determine the type of sand within this sample size. Figure 1(b) shows an optical image of the polished sand surface with grains oriented in different directions. The sand particle size distributions express the percentage (by mass) of individual size ranges [24]. Results and Discussion Young’s Modulus and Hardness For the measurement of elastic modulus and hardness. To determine whether the sand grains are isotropic or not. This indenter can reach a maximum indentation depth of 500 μm and a maximum load of 500 mN.2 nm and 50 nN. Nanoindentations were made on flat. The sieves use metal wire cloth with an ASTME-11 standard sieve series. H¼ Pmax Ac ð1Þ where Pmax is the maximum indentation force and Ac is the contact area corresponding to the contact depth (hc) at maximum load. Hence. a given force was applied by the nanoindenter using a Berkovich tip on the sand grain. The displacement and load resolutions are 0. in which the indenter-tip was pressed onto the sample and the load on it was increased linearly. defect structure. a module of the MTS Nanoindenter system. while Ei and νi are the Young’s modulus and Poisson’s ratio of the indenter tip (made of diamond). crystal orientation. its modulus has to be considered in the expression for the calculation of specimen modulus from the contact stiffness. the continuum approximation can be applied and equations (1) to (3) can be used to determine the modulus and hardness of the samples. as the main objective of this investigation was to determine the mechanical properties of individual grains by nanoindentation. K and Fe in the sand sample. polished sand grain surfaces under a constant rate of loading. the holding time in our experiments was zero. Multidirectional shaker was used to determine the size distribution of the sand grains. The sand grains exhibited different colors due to differences in the density. At this point. Both Berkovich and cubecorner indenter tips. Within the sand grains. and modulus and Poisson’s ratio values for the indenter tip can be used to determine the elastic modulus for a specimen corresponding to its Poisson’s ratio. Due to finite stiffness of the indenter tip. The applied load on the indenter tip was increased until it reached a user-defined value. S¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi dP 2 ¼ pffiffiffi Er Ac ðhc Þ dh p ð3Þ Equations (2) and (3) along with the known values of the area function. the sand that falls through a mesh was given the designation of passed weight and the sand that remains on top of that mesh was designated as remaining weight. The samples were cured in an oven resulting in a composite of sand grains in a hardened epoxy matrix. Hardness (H) is obtained using. the load is held constant for a period or removed. respectively. Ca. To separate different sizes of the sand. The grain size distribution used in this investigation was 0. slope of the unloading curve. no information is extracted from the hold segment. made of single crystal diamond. Figure 2(b) shows a typical nanoindentation load-displacement curve. Nanoindentations were carried out under load-control. Since. Thus. The reduced modulus Er is obtained using 1 1 À n2 1 À n2 s i ¼ þ . the sand agglomerate was vibrated through a series of progressively smaller sieves that were stacked one on top of another. respectively. Figure 3(a) and (b) show Weibull plots for the data obtained on Young’s modulus and hardness.08±5. and Z-cut surfaces. were found to be lower. 2 (color online) Nanoindentation test on sand grain using Berkovich indenter. This could be induced by different material constituents in the sand grain. face YZ.9 GPa and 10.7 GPa (range 5. These were direct outputs from the nanoindentation software. 79 GPa.Exp Mech Fig.74 GPa. nanoindentations were carried out on 250 different grains with two indentations per grain. which use . respectively. and 44. Y-cut. however. (a) Residual indent impression and 3D inverse image. On each face. respectively. compared to the modulus of quartz.1 GPa (range 41. 45. Young’s modulus values. respectively.21 GPa. The results from the two tests were averaged and taken as the mean value for that grain.7 GPa). The Weibull distributions of the modulus and the hardness values were plotted in [Fig.85 GPa.7 mm). we determined the median value (corresponding to P50 value) for Young’s modulus of the sand to be 90. indicating that the sand grain can be modeled as an isotropic material.7 mm).4 to 13.8 GPa) and hardness to be 10. The three faces were designated as: face XY. Nanoindentation tests were made on all three faces. and face ZX of the sand grain.6±3. Inverse methodologies. Using Weibull distribution function. 4(a) and (b)]. Once an estimate was obtained for a single grain (of representative size ∼0. and 103 GPa when indentations were made on an X-cut.30 GPa. Young’s modulus and hardness values were obtained using the slope of the unloading curve and the values of load and contact area at maximum indentation depth. nanoindentations were made on a sand grain with the same maximum load at different locations. The Young’s modulus values for face XY. 8 to 10 tests were carried out and the distances between the neighboring test locations were kept apart by at least 50 μm. Scatter in the data is attributed to different types of sand grains due primarily to the variations in the material constituents. and ZX were 43. In order to investigate the variability in the properties over a single grain (size ∼0. which has modulus values of 79 GPa.8 mm×0. YZ. These values are very close to each other.4 to 115.7 mm) embedded in an epoxy matrix. and crystal orientations. defect structure.7±4. The variation in the measured modulus and hardness at different locations is attributed here to the heterogeneity in the sand grain. (b) Loaddisplacement curve orthogonal directions on the cube corner of a sand grain (1 mm×0. respectively. respectively. The median value (corresponding to P50 probability) for Young’s modulus and hardness are 66. Stress-Strain Response Solution for the indentation problem has been well established [25–30]. 4 (a) Weibull plot for Young’s modulus from nanoindentations on different sand grains. (b) Weibull plot of hardness from nanoindentations on different sand grains . by comparing the load-displacement curves.g.g. 3 (a) Weibull plot of Young’s modulus from nanoindentations on a single sand grain. Á pffiffiffiffiffiffiffiffiffi À " ¼ "ij "ij ¼ "y þ "p . The von Mises equivalent stress is given as. 2 Fig. such as Finite Element Method (FEM). nanoindentation was modeled using FEM. It maybe noted that FEM has been applied successfully in the simulation of nanoindentation problem [31–40]. For determining the stress-strain relationship of granular sand material. To predict the elastic-plastic properties. In such cases. K is the reference stress value. are used in situations where it is difficult to extract mechanical properties using analytical solutions alone due to nonlinearities or complexities in the material. in the case of nanoindentation. (b) Weibull plot of hardness from nanoindentations on a single sand grain experimentation in combination with numerical techniques to aid in characterizing the material properties. it is easier to simulate the equivalent model using a numerical method. plastic flow. where εy is the yield strain and εp is the plastic qffiffiffiffiffiffiffiffiffiffiffiffiffi strain. The plastic behavior under compression was assumed to satisfy the RambergOsgood relationship between the true stress and true strain as s 1=n "¼ for s ! s y ð4Þ K where n is the work hardening exponent. s ¼ 3 s ij s ij . The values are adjusted so that numerical results obtained can be comparable with the experimental values. for e. von Mises yield criterion was used along with isotropic hardening to simulate the deformation characteristics of a sand grain. geometry and loading conditions. Initially.Exp Mech Fig. we can start with certain assumed values of material parameters for e. and ε is the equivalent strain. ABAQUS V6. Figure 5 shows the fit obtained using the FEM simulation (obtained using ABAQUSStandard) to the representative experimental data of P50. for s sy ð5Þ where E is the Young’s modulus. Poisson’s ratio of 0.8-4 standard [41] software program was used to perform the calculations. and σ3 are the principal stresses and σy is the yield strength measured in a uniaxial tension test. 5 Comparison of load-displacement relationship curves from nanoindentation and FE simulation Within the elastic limit. the continuum plasticity material model was justified. Thus. the maximum strain up to which the stress-strain curve is valid. It should be noted that using this approach.17 and Young’s Modulus of 75 GPa were used in this simulation. No cracks were observed by examining the indent impressions (for the Berkovich indenter tip) obtained from NanoVision. Because of the Berkovich indenter’s axisymmetric geometry. The mesh size selected was tested for convergence of the loaddisplacement curve. In this simulation. As stated earlier. the material parameters were adjusted until a good agreement was reached between experimental and numerical data. 6 (a) Compressive strain distribution along radial direction of the model. Thus. This numerical load was plotted versus the displacement into the surface to give load-displacement curve from the simulation. The best-fit parameters were then used to determine the stressstrain relationship for the sand grain sample. In order to obtain a better fit. The von Mises yield criteria can be written as ðs 1 À s 2 Þ2 þ ðs 2 À s 3 Þ2 þ ðs 3 À s 1 Þ2 ¼ 2s 2 y ð6Þ where. The output of the FE analysis was the resulting reaction force or load. The displacement history from the experiment was given as input for the FE analysis. we used an inverse problem solving approach to determine the stressstrain relationship of sand at granular level by correlating the FEM simulated nanoindentation load-displacement data with the measured results. the tip of the Berkovich indenter was assumed to be perfectly sharp but in practice. only one sixth of the entire model was used in this simulation to reduce the computational time. σ1. assuming finite deformation characteristics. s ¼ E ".Exp Mech Fig. is limited by the strain produced by the nanoindentation test [42]. (b) Stress-strain curve (in compression) from FE simulation . σ2. The Berkovich indenter was simulated based on its three sided pyramidal geometry. Finite Element (FE) method was used in 3D simulations of nanoindentation. The perfectly sharp tip model in the simu- Fig. The numerical values of the loaddisplacement curve were compared with the experimental values and the measure of fit was carried out by minimizing the least squares correlation coefficient. the indenter has a tip with a finite radius of tens to a few hundred nanometers. the initially assumed values were so adjusted as to minimize the least squares correlation coefficient. decrease slowly from 0. 8 Weibull plot of fracture toughness from nanoindentations on different sand grains Fig. We consider a strain of 0. 6(a)]. Since an artificially sharp Berkovich indenter model was used in the simulation. At a depth less than 200 nm. (b) at maximum load of 70 mN lations will cause the stress near the tip to be much higher.Exp Mech Fig. 6(b)] is represented in the form Fig. starting from the tip contact area is shown. the compressive strain distribution along the radial direction. it will result in unreasonable values near the tip. The predicted stress-strain response [Fig. In [Fig. The strains in the region at a radial distance of more than 200 nm from the tip. 7 (color online) Surface profile showing cracks generated after nanoindentation using cube corner tip.6. 9 (color online) Inverse image of nanoindentation on a sand grain at 5 mN load using a cube corner nanoindenter tip . as the highest strain.6. (a) at maximum load of 80 mN. which is in the region 200 nm away from the tip. the strains are unreasonably high (∼90% compressive strain) at the tip due to singularity. Further studies are needed to confirm this hypothesis. Before proceeding with the nanoindentation on sand grains. Lade PV.6 MPa-m0. Eglin AFB. the variation in the property value indicates that sand at granular level is very inhomogeneous due to different material constituents. Endowed Chair in Engineering for additional financial support. This may be the minimal load below which the sand grains would behave in a ductile manner with a possible ‘size effect’ implications. [22] was used.4 to 13. Holubec I. J Soil Mech Found Div 99(10):793–812 . radial cracks propagate out of the indenter corners. and stress-strain relationship. Nelson. H. It maybe noted that cracks observed on some sand grains were not exactly straight. 7(a) and (b)]. such as presence of References 1.8 to 3.Exp Mech of Ramberg-Osgood model and the parameters obtained were n=0. The average value for fracture toughness obtained was 0. William L Cooper of the Air Force Research Laboratory. a formula derived by Antis et al.25 and K=14.1 GPa (range 41.6 MPa-m0. and crystal orientations. The Weibull distribution of the fracture toughness values obtained is shown in Fig. Conclusions In order to assess the granular-level mechanical behavior of sand.7 GPa (range 5. This is attributed to the possible inhomogeneity within a single sand grain. The data reported here can be used for mesoscale (granular) simulations of sand in which the individual sand grains would have different properties along with a range of distributions obtained in this study. Victor Giurgiutiu of the Air Force Office of Scientific Research (AFOSR). the median value for the fracture toughness was obtained as 1. In order to investigate the failure behavior of the sand grains.032 [23]). In such a case.L. Can Geotech J 3 (4):179–190 2. defect structure.77 MPa-m0. acknowledges the A. acknowledges the support of NSF (CMMI-0555902 and CMS-9985060) and R.4 to 115.5). From it.77 MPa-m0. The authors also thank Mrs. the crack length c is determined from the surface scans (two of the scans along with inverse images) as shown in [Fig. we obtained an average value of the ratio as 8. Sherbourne AN (1966) Yielding and flow of sand in triaxial compression Part I. Since we are interested in the ratio of E/H from the nanoindentation tests for calculation of fracture toughness. namely. The yield stress obtained from the simulation results is σy =6. It is well known that sand grains would behave in a brittle manner under load. where c is the crack length. Power-law was used to represent the homogenized and isotropic stressstrain behavior of sand at the granular level.8 GPa).K. Duncan JM (1973) Cubical triaxial tests on cohesionless soil. fracture toughness. It maybe noted that due to slight errors in the alignment and inhomogeniety of the sand grains. a wide variation in the granular behavior of sand was observed. Acknowledgements This work is supported by an AFOSR DEPSCoR grant (FA9550-08-1-0328) and the authors acknowledge the strong support and interest of Dr. the values of the elastic modulus (E) and hardness (H) were obtained initially using the Berkovich nanoindenter tip. and Dr. KC ¼ a 0:5 E P H c3=2 ð7Þ defects. α is an empirical constant that takes into account the geometry of the indenter tip (for a cube-corner tip α=0. In equation (7).7 GPa). a cube corner indenter was used to indent into a polished sand grain to initiate cracks at the corners of three edges of the tip.27 GPa. Need exists to establish the link between nanoindentation measured properties of sand grains with the observed macro properties for accurate prediction of the bulk behavior of sand. and fracture toughness to be 1. This was followed by indentations with a cube-corner tip (see “Young’s Modulus and Hardness”). Representative values (50 percentile) of Young’s modulus (P50) for the sand grains was found to be 90.8 to 3. [43]. In order to estimate the fracture toughness. nanoindentation tests were conducted on individual sand grains to characterize their mechanical properties. When there are differences between the crack lengths from the same indent.6 MPa-m0. hardness to be 10. Both 3D inverse images and 2D nanoindentation residual images are shown for details on the crack formation and fracture.5 which is in reasonable agreement with the value reported for fused silica by Harding et al.1 GPa. Fracture Toughness When a brittle material is loaded by a sharp indenter under an appropriate load. it is somewhat difficult to induce three cracks with equal length. Sharon Green for her highly skilled editorial assistance. 8.5 (range 0.5). However. Young’s modulus. For sand. As can be expected of a brittle material. 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