Sources of error in relating nanoindentation results to material properties
This dissertation consists of four submission-ready papers that address some of the key error sources that affect the accuracy of interpretation of nanoindentation test results to obtain material properties for elastoplastic materials. The first part of the work is a study of the effect of sample tilt on results of nanoindentation tests. Geometrical relations are used to develop a correction to account for the effect of tilt angle on the contact area. 3D FEA (Finite Element Analysis) shows that the assumptions made in deriving the geometric correction are valid, and the results for contact area, hardness and modulus match the predictions of the analytical model. It is shown that for both materials that sink-in and those that pile-up, the projected contact area for nanoindentation on tilted sample is higher than that estimated by the standard area function, which leads to overestimation of the hardness and elastic modulus. Experimental nanoindentation tests on tilted samples show lower sensitivity to sample tilt compared to FEA results because the compliance of the indenter holder causes the indenter tip to displace in the direction of the surface tilt, reducing the total penetration of the tip into the surface. For tips with very high compliance, this may even lead to significant underestimation of the hardness and modulus. The second part discusses the various factors that affect the accuracy of FEA of nanoindentation. With the understanding that contact area error arising from discretization of the continuum is a key contributor to noise in the hardness data, a self similar mesh is designed that results in a known amount of maximum error in contact area over a range of depths of penetration of the indenter. Based on the fact that contact area increases in discrete jumps, it is argued that the maximum force that a given area of contact can support, before the next element comes into contact, is the best measure of the true hardness of the material that can be obtained with a given mesh. FEA simulations carried out with meshes of different amounts of error in contact area show that as the discretization becomes coarser, the estimate of the true hardness increases, due to the inability of the mesh to resolve the steep gradients in stress and strain near the end point of contact. It is also shown that results obtained from different meshes with different error percentages can be extrapolated to determine the exact value of hardness that will be obtained with infinitesimally small elements. It is shown that other sources of error, such as the convergence tolerance of the iterative solution process, are small in comparison to the discretization errors. The third part is a study aimed at identifying the size of the volume underneath a nanoindentation that influences the hardness and modulus measured. FEA simulations of the indentation of a hemispherical particle embedded in a matrix reveal that the hardness of particle can be measured accurately by nanoindentation as long as vii the plastically deformed region is confined entirely within the particle. While this may be intuitively obvious in retrospect, this is the first quantitative demonstration that this is so. It is found that an available relationship between the force, yield stress, and the radius of the plastically deformed zone is accurate under the conditions studied. This can be used to determine the maximum penetration depth that can be used if the size of the particle is to be estimated. For modulus of elasticity, it is shown that the modulus measured by nanoindentation method actually represents the elastic response of the entire specimen at the indentation point, which for all penetration depths, is a composite of the elastic response of both the particle and the matrix. A relationship is developed that shows the effect of boundary conditions and the matrix on the modulus measured by indentation at low depths of penetration for a hemispherical particle/matrix system. The last part describes a new iterative procedure for estimation of the mechanical properties of elasticperfectly plastic materials by nanoindentation. The key feature of this method is the estimation of the correct contact height, irrespective of whether the material piles-up or sinks-in, using an iterative procedure. It is shown that the proposed method improves the estimation of hardness and modulus compared to the Oliver and Pharr method and also gives a good estimation of the yield stress for materials with plastic index greater than 10.
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