The indentation load-displacement behavior of six materials tested with a Berkovich indenter has been carefully documented to establish an improved method for determining hardness and elastic modulus from indentation load-displacement data. The materials included fused silica, soda–lime glass, and single crystals of aluminum, tungsten, quartz, and sapphire. It is shown that the load–displacement curves during unloading in these materials are not linear, even in the initial stages, thereby suggesting that the flat punch approximation used so often in the analysis of unloading data is not entirely adequate. An analysis technique is presented that accounts for the curvature in the unloading data and provides a physically justifiable procedure for determining the depth which should be used in conjunction with the indenter shape function to establish the contact area at peak load. The hardnesses and elastic moduli of the six materials are computed using the analysis procedure and compared with values determined by independent means to assess the accuracy of the method. The results show that with good technique, moduli can be measured to within 5%.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A53D45298E14F73B86495BBC056E16DC/S0884291400017039a.pdf/div-class-title-an-improved-technique-for-determining-hardness-and-elastic-modulus-using-load-and-displacement-sensing-indentation-experiments-div.pdf

An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments

The method we introduced in 1992 for measuring hardness and elastic modulus by instrumented indentation techniques has widely been adopted and used in the characterization of small-scale mechanical behavior. Since its original development, the method has undergone numerous refinements and changes brought about by improvements to testing equipment and techniques as well as from advances in our understanding of the mechanics of elastic–plastic contact. Here, we review our current understanding of the mechanics governing elastic–plastic indentation as they pertain to load and depth-sensing indentation testing of monolithic materials and provide an update of how we now implement the method to make the most accurate mechanical property measurements. The limitations of the method are also discussed.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0884291400085800

Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology

One of the simplest, and most completely treated, fields of application of quantum mechanics is the theory of atoms with one or two electrons For hydrogen and the analcgous ions He+, Li++, etc, the calculations can be performed exactly, both in Schrodinger’s nonrelativistic wave mechanics and in Dirac’s relativistic theory of the electron More specifically, the calculations are exact for a single electron in a fixed Coulomb potential Hydrogen-like atoms thus furnish an excellent way of testing the validity of quantum mechanics For such atoms the correction terms due to the motion and structure of atomic nuclei and due to quantum electrodynamic effects are small and can be calculated with high accuracy Since the energy levels of hydrogen and similar atoms can be investigated experimentally to an astounding degree of accuracy, some accurate tests of the validity of quantum electrodynamics are also possible Finally, the theory of such atoms in an external electric or magnetic field has also been developed in detail and compared with experiment

Quantum Mechanics of One- and Two-Electron Atoms

Publisher Summary In recent years, the interest in the problem of brittle fracture and, in particular, in the theory of cracks has grown appreciably in connection with various technical applications. Numerous investigations have been carried out, enlarging in essential points the classical concepts of cracks and methods of analysis. The qualitative features of the problems of cracks, associated with their peculiar nonlinearity as revealed in these investigations, makes the theory of cracks stand out distinctly from the whole range of problems in terms of the theory of elasticity. The chapter presents a unified view of the way basic problems in the theory of equilibrium cracks are formulated and discusses the results obtained thereby. The object of the theory of equilibrium cracks is the study of the equilibrium of solids in the presence of cracks. However, there exists a fundamental distinction between these two problems, The form of a cavity undergoes only slight changes even under a considerable variation in the load acting on a body, while the cracks whose surface also constitutes a part of the body boundary can expand even with small increase of the load to which the body is subjected.

The mathematical theory of equilibrium cracks in brittle fracture

http://experimentationlab.berkeley.edu/sites/default/files/AFMImages/butt_AFM.pdf

Force measurements with the atomic force microscope: Technique, interpretation and applications

The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile

The use of integral transforms

The distribution of stress produced in the interior of an elastic solid by the opening of an internal crack under the action of pressure applied to its surface is considered. The analysis is given for 'Griffith' cracks (section 2) and for circular cracks (section 3), it being assumed in the latter case that the applied pressure varies over the surface of the crack. For both types of crack the case in which the pressure is constant over the entire crack surface is considered in some detail, the stress components being tabulated and the distribution of stress shown graphically. The effect of a crack (of either type) on the stress produced in an elastic body by a uniform tensile stress is considered and the conditions for rupture deduced.

Ian N. Sneddon

Papers

The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile

The use of integral transforms

The Distribution of Stress in the Neighbourhood of a Crack in an Elastic Solid

Progress in solid mechanics

Mixed boundary value problems in potential theory