Per-Lennart Larsson

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.

Analysis of Berkovich indentation

The Berkovich indentation test is analysed numerically, using the finite element method, and experimentally. The results derived are pertinent to indentation of elastic materials and metals and include universal formulae for the load-indentation depth relation and the hardness, as well as a detailed study of the mechanical fields involved at loading and unloading. Large strain elastic and elastoplastic results are compared with small strain ones and similarities, as well as differences, are discussed in some detail. Special attention is given to a comparison between the characteristics of Berkovich indentation and the Vickers hardness test. The accuracy of relevant formulae for determining the elastic stiffness during the unloading process is checked. Experiments are performed both on the nano- and microscale. Numerical and experimental findings are compared in detail, especially as regards bulk results.

On the determination of residual stress and strain fields by sharp indentation testing.: Part I: theoretical and numerical analysis

Abstract Sharp indentation tests, presently represented by cone and Vickers indenters, are analysed theoretically and numerically in order to explore how equi-biaxial residual stress and strain fields can be determined from the global properties, i.e. the size of the contact area between indenter and material and the hardness, given by such tests. It is shown that the residual strain fields can be accurately correlated with the hardness value while residual stresses are related to the size of the contact area. The latter feature is explained by the fact that the size of the contact area is sensitive to elastic effects. The results are summarized in simple closed form relations, well-suited to be used in an experimental situation, and the range of validity for the resulting formulae is discussed. The predictions are compared with corresponding results taken from the literature and good agreement is found. An experimental scheme for determination of residual fields by indentation is also suggested.

Similarity analysis of inelastic contact

Analysis of mechanical contact of solids is of interest not only regarding a variety of mechanical assemblies but also on a smaller scale such as roughness properties of surfaces and compaction of powder particles. Indentation testing is another prominent problem in the context. To analyse the phenomena involved is inherently difficult at application essentially due to the presence of large strains, nonlinear material behaviour, time dependence and moving contact boundaries. Recently, progress has been made, however, to explicitly solve basic boundary value problems especially due to advances in computational techniques. A substantial ingredient which facilitates solution procedures is self-similarity and it is the present purpose to explore in detail the advantages in a general setting when this feature prevails. A viscoplastic framework is laid down for a wide class of constitutive properties where strain-hardening plasticity, creep and also nonlinear elasticity arise as special cases. It is then shown that when surface shapes and material properties are modelled by homogeneous functions, associated boundary value problems posed may be reduced to stationary ones. As a consequence, within Hertzian kinematics, relations between contact impression and regions become independent of loading and time and the connection to loading characteristics does not usually require a full solution of the problem. In particular it is shown that for general head-shapes it proves efficient to use an approach where an intermediate flat die solution serves as a basic tool also for hereditary materials. An invariant computational procedure based on the intermediate problem is arrived at and decisive results shown to be found by simple cumulative superposition. Illustrations are given analytically for ellipsoidal contact of Newtonian fluids and by detailed computations for spherical indentation of viscoplastic solids for which also universal hardness formulae are proposed. For several bodies in contact it is shown how general results may be extracted from fundamental solutions for a half-space.

On Brinell and Boussinesq indentation of creeping solids

As an alternative to traditional tensile testing of materials subjected to creep, indentation testing is examined. Axisymmetric punches of shapes defined by smooth homogeneous functions are analysed in general at power law behaviour both from a theoretical and a computational point of view. It is first shown that by correspondence to nonlinear elasticity and self-similarity the problem to determine time-dependent properties admits reduction to a stationary one. Specifically it is proved that the creep rate problem posed depends only on the resulting contact area but not on specific punch profiles. As a consequence the relation between indentation depth and contact area is history independent. So interpreted, the solution for a flat circular cylinder (Boussinesq) is not only of intrinsic interest but serves as a reference solution to generate results for various punch profiles. This is conveniently carried out by cumulative superposition and in particular ball indentation (Brinell) is analysed in depth. A carefully designed finite element procedure based on a mixed variational principle is used to provide a variety of explicit results of high accuracy pertaining to stress and deformation fields. Universal relations for hardness at creep are proposed for Boussinesq and Brinell indentation in analogy with the celebrated formula by Tabor for indentation of strain-hardening plastic materials. Quantitative comparison is made with a diversity of experimental data attained by earlier writers and the relative merits of indentation strategies are discussed.

Analysis of pyramid indentation of pressure-sensitive hard metals and ceramics

Abstract In the last two decades, pyramid micro and nano-indentation tests such as Vickers (tetragonal base), Berkovich (trigonal base), and Knoop (rhomboid base) have been used on pressure sensitive materials, including hard metals, glasses and ceramics, and were found to give valuable mechanical and other physical information which may be otherwise difficult to obtain. Such indentation experiments are attractive because they require only a small flat area of specimen and are relatively easy to perform on materials of high stiffness and brittleness. More important, at least in principle, a wide load range can be used, so that many mechanical properties can be inferred by recording continuously the force-depth response at loading and unloading. However, the numerous reports that exist on this topic lack detailed and rigid analysis, with current analytical methods relying heavily on simplifications and speculations about the stress fields and deformation patterns. The indentation force-depth relation, the imprint morphology (e.g., sinking-in, cracking, etc.) and the microscopic observation of the regions around and beneath the imprint are the main direct experimental results from indentation tests. To explore the mechanical information included in the experimental observations of pyramid indentations, we performed a parametric analysis of Vickers and Berkovich indentations using the finite element method. The pressure sensitivity of the materials was modeled according to the classic Drucker-Prager plastic potential. The detailed analysis of the numerical results enabled us to explain much of the phenomenology of standard pyramid indentation tests on metals and ceramics.

On the determination of residual stress and strain fields by sharp indentation testing.: Part II: experimental investigation

An experimental investigation has been carried out in order to study how residual stress and strain fields can be determined by sharp indentation testing. Aiming at a thorough understanding of the influence from general residual surface stress and strain fields on global indentation parameters, i.e. hardness and contact area, Vickers indentation tests have been performed on specimens first exposed to standard four-point bend (4PB) or single edge notch bend (SEN(B)) loading. The experimental results have been evaluated based on the findings in a parallel theoretical/numerical investigation and are compared with finite element simulations as well as with corresponding results taken from the literature. Good agreement between experiments and numerical results has been found, particularly in a situation with no or little plastic deformation due to preloading.

Investigation of sharp contact at rigid-plastic conditions

Sharp contact problems are examined theoretically and numerically. The analysis is focused on elastic-plastic material behaviour and in particular the case when the local plastic zone arising at contact is so large that elastic effects on the mean contact pressure will be small or negligible. It is shown that, save for the particular case of a rigid-plastic power-law material, at such conditions, there is no single representative value on the uniaxial stress-strain curve that can be used in order to evaluate the global parameters at contact. However, the present numerical results indicate that good accuracy predictions for the mean contact pressure can be achieved when this variable is described by two parameters corresponding to the stress levels at, approximately, 2 and 35% plastic strain. Regarding the size of the contact area, it is shown that this quantity is very sensitive to elastic effects and any general correlation with material properties is complicated at best. The numerical analysis is performed by using the finite element method and the theoretical as well as the numerical results are compared with relevant experimental ones taken from the literature. From a practical point of view, the presented results are directly applicable to material characterization or measurements of residual mechanical fields by sharp indentation tests, but also for situations such as contact in gears or in electronic devices.

On microindentation of viscoelastic polymers

Indentation testing as a tool for determination of the mechanical properties of viscoelastic polymers is examined in some detail. Both ball (Brinell) and flat punch (Boussinesq) indentation are analysed theoretically and simple but rigorous formulae for a complete characterization of linear materials are presented. In particular, for the Brinell test the relation between actual contact area and the size of the residual imprint left on the surface after unloading is analysed using the finite element method. Indentation experiments are performed for two polymers of commercial interest. The mechanical properties determined experimentally by indentation are compared with corresponding results from standard uniaxial relaxation tests, and good agreement was found.

Analysis of cold and hot isostatic compaction of spherical particles

Cold and hot isostatic compaction of monosized metal powders is analysed within the framework of viscoplastic theory for particles packed both in a regular and a random manner. The problem of local contact between spherical particles is analysed first by drawing upon recent detailed numerical studies of spherical indentation of power law plastic and creeping solids. The compaction process is then modelled as a self-similar contraction of unit (average) cells, in contrast to commonly used phenomenological assumptions, yielding a simple but rigorous relation between the densification density and centre-to-centre approach of adjacent powder particles. The resulting densification formulae are easy to apply and the influence of hardening and creep parameters appears in concise form. Predictions are compared with discriminating experimental data for a variety of different powder materials, both in the cold and a hot state, and the agreement is good for a relative density increase of up to 30%.