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

We provide an overview of the basic concepts of scaling and dimensional analysis, followed by a review of some of the recent work on applying these concepts to modeling instrumented indentation measurements. Specifically, we examine conical and pyramidal indentation in elastic-plastic solids with power-law work-hardening, in power-law creep solids, and in linear viscoelastic materials. We show that the scaling approach to indentation modeling provides new insights into several basic questions in instrumented indentation, including, what information is contained in the indentation load-displacement curves? How does hardness depend on the mechanical properties and indenter geometry? What are the factors determining piling-up and sinking-in of surface profiles around indents? Can stress-strain relationships be obtained from indentation load-displacement curves? How to measure time dependent mechanical properties from indentation? How to detect or confirm indentation size effects? The scaling approach also helps organize knowledge and provides a framework for bridging micro- and macroscales. We hope that this review will accomplish two purposes: (1) introducing the basic concepts of scaling and dimensional analysis to materials scientists and engineers, and (2) providing a better understanding of instrumented indentation measurements.

Scaling, dimensional analysis, and indentation measurements

Mechanical models for living cells--a review

In this review, we compare the reported values of Young's modulus (YM) obtained from indentation and tensile deformations of soft biological tissues. When the method of deformation is ignored, YM values for any given tissue typically span several orders of magnitude. If the method of deformation is considered, then a consistent and less ambiguous result emerges. On average, YM values for soft tissues are consistently lower when obtained by indentation deformations. We discuss the implications and potential impact of this finding.

http://chemotaxis.semmelweis.hu/CHTXhpg/Durotaxis/DT2/McKeeCT_TissueEng_2011.pdf

Indentation versus tensile measurements of Young's modulus for soft biological tissues.

During the course of some investigations on the distribution of stress in an elastic solid it was noticed by the senior author that a systematic application of the method of integral transforms to the problem of the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch reduced the problem essentially to one of solving a pair of integral equations belonging to a class which has been studied by Titchmarsh and by Busbridge (4,5). This procedure allows one to obtain the solution for an arbitrary shape of punch by a general method which leads automatically to the solution and avoids the troublesome procedure, adopted by Love (2) in the case of a conical punch, of being obliged to guess appropriate combinations of solutions which will satisfy the prescribed boundary conditions in any special case. Moreover, it can easily be seen that an attempt to apply Love's method to more complicated shapes of punch will lead to considerable analytical difficulties.

I. N. Sneddon

Papers

The elastic stresses produced by the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch