Numerical investigations of pyramid indentation on powder compacts

Finite element modelling and simulation of indentation testing: a bibliography (1990‐2002)

Abstract: This paper gives a bibliographical review of the finite element modelling and simulation of indentation testing from the theoretical as well as practical points of view. The bibliography lists references to papers, conference proceedings and theses/dissertations that were published between 1990 and 2002. At the end of this paper, 509 references are listed dealing with subjects such as, fundamental relations and modelling in indentation testing, identification of mechanical properties for specific materials, fracture mechanics problems in indentation, scaling relationship for indentation, indenter geometry and indentation testing.

Quasi-static normal indentation of an elasto-plastic substrate by a periodic array of elastic strip punches

Abstract: The plane-strain, elasto-plastic, contact problem described in the title is treated for a substrate that is perfectly plastic post yield, and so simulates compression molding of some metals at elevated temperatures. The analysis uses finite elements and is verified with test problems and convergence checks. The key finding is that, in what might reasonably be viewed as a fully-plastic state, the molding pressure normalized by the yield stress is equal to a constant plus a term that increases linearly with the depth of indentation. This is in contrast to Tabor’s classical result for hardness testing that has the normalized pressure solely equal to a constant when response is fully plastic. The additional linear stiffening term found with the finite element analysis of the present configuration is confirmed experimentally. An explanation of the source of this stiffening term even with a perfectly-plastic substrate is offered. Contact stresses are also tracked as indentation proceeds. These stresses initially have high stress concentrations near the edges of the strip punches. However, these peak stresses abate rapidly with plastic flow and approach a nearly uniform distribution within the fully-plastic state. Implications for compression molding are discussed.

Sources of error in relating nanoindentation results to material properties

Abstract: 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…

The correlation of indentation experiments

Abstract: The theory of rigid perfectly-plastic solids predicts indentation pressures, using wedge-shaped or conical indenters, which depend only on the geometry of the indenter and the yield stress of the material. With blunt wedges or with materials having a low ratio of Young's modulus, E, to yield stress, Y, the material displaced by the indenter is accommodated by an approximately radial expansion of the surrounding material. The indentation pressure then falls below the rigid perfectly-plastic value. In these circumstances, measurements of indentation pressure for a variety of indenter geometries are shown to correlate with the single parameter (E/Y) tan β, where β is the angle of inclination of the indenter to the surface at the edge of the indentation. This parameter may be interpreted as the ratio of the strain imposed by the indenter to the yield strain of the material. A simplified theoretical model of this behaviour is obtained by extending R. Hill's theory of expanding a cylindrical or spherical cavity in an elastic-plastic material to ensure compatibility between the volume of material displaced by the indenter and that accommodated by elastic expansion.

Elastic analysis of some punch problems for a layered medium

Abstract: The problems of flat-ended cylindrical, quadrilateral, and triangular punches indenting a layered isotropic elastic half-space are considered. The former two are analyzed using a basis function technique, while the latter problem is analyzed via a singular integral equation. Solutions are obtained numerically. Load-deflection relations are obtained for a series of values of the ratio of Young's modulus in the layer and substrate, and for a variety of punch sizes. These solutions provide an accurate basis for the estimation of Young's modulus of thin films from the initial unloading compliance observed in indentation tests, and are specifically relevant to axisymmetric, Vicker's, and triangular indenters. The results should also be of interest in foundation engineering.

Analysis of Vickers indentation

Abstract: Indentation tests have for a long time been a standard method for material characterization as they provide an easy, inexpensive, non-destructive and objective method of evaluating basic properties from small volumes of materials. Besides hardness, recently also aspects of, for example, toughness and residual stresses have been advantageously investigated by indentation. Sharp indentation tests such as Vickers, Berkovich and Knoop lack however a solid mechanical foundation. The present paper aims mainly to explore the theoretical foundation for the commonly used Vickers test. The investigated types of constitutive behavior include isotropic linear elasticity and plasticity. The influence of large elastoplastic deformations was also assessed. Extensive computation was required, based on the finite element method. In addition, the analysis was compared with depth-sensing indentation experiments. The results put forward explanations of hardness formulae for many standard materials such as metals, as well as the stress and deformation analysis relating to the Vickers indentation test.

Finite element simulation of indentation experiments

Abstract: Indentation experiments are now being used to study the elastic and plastic properties of materials on a sub-micrometer scale. Simulations of these experiments have been performed using the finite element method under the conditions of frictionless and completely adhesive contact and within the context of incremental elasto-plasticity. Comparison of these simulated results with experimental results demonstrates that the continuum based finite element approach has the capability to determine the load depth response of a sub-micrometer indentation test. It is also shown that the hardness and elastic modules of the can be obtained from the loading and unloading portoins of these simulated curves.

Void growth and failure in notched bars

Abstract: Void growth and ductile failure in the nonuniform multiaxial stress fields of notched bars are studied numerically and experimentally. U-notched bars with different notch acuities are made from partially consolidated and sintered iron powder compacts with various residual porosities. The materials are modelled using an elastic-viscoplastic constitutive relation that accounts for strength degradation resulting from the growth of microvoids. The matrix stress-strain relation and the initial void volume fractions used in the calculations are determined experimentally. The remaining parameters in the constitutive equations are evaluated from micromechanical models. Comparisons of the calculations with experimental results indicate that the constitutive model can provide good estimates of the evolution of the void volume fraction and of the strength reduction induced by void growth under a variety of nonuniform stress histories.