Nanoindentation – The Significance of Hertzian Contact and Development of Plastic Zones

Nanoindentation is a method of measurement of the mechanical properties of small volumes of materials using an instrumented indentation technique. Elastic modulus, hardness, fracture toughness, creep and dynamic properties such as storage and loss moduli can be measured. In this and subsequent articles, we will look at some of the issues facing the user of a nanoindentation instrument. Our purpose is to educate and inform the prospective user of this type of equipment as to what can be measured and what factors influence the results obtained.

The IBIS Nanoindentation system from Fischer-Cripps Laboratories.

Figure 1. The IBIS Nanoindentation system from Fischer-Cripps Laboratories.

Spherical Indenters, Flat Specimens and Hertzian Contact

The nature of the contact between a spherical indenter and a flat surface was first studied in detail by Hertz in the 19th century. This has significance for us in nanoindentation testing because the initial contact with the indenter takes the form of a spherical indenter on a flat specimen surface, even for a sharp tip, because all real tips have a radius and are not atomically sharp. Under these conditions, the initial contact is usually elastic because the level of shear stress in the Herztian stress field is insufficient to cause plastic deformation in the specimen material. Hertzian equations of contact show that when there is no plasticity, and the contact is elastic, the mean contact pressure pm (the indentation stress) is linearly proportional to the ratio a/R which is called the indentation strain.

Indentation stress and indentation strain

Indentation stress and indentation strain

Figure 2. Indentation stress and indentation strain

Shear Stresses in Specimens

As the load is increased, the shear stress in the specimen increases. The principal shear stress for indentation with a spherical indenter is a maximum at 0.47pm at a depth of 0.5a beneath the specimen surface directly beneath the indenter . We may employ either the Tresca or von Mises shear stress criteria, where plastic flow occurs at t » 0.5Y, to show that plastic deformation in the specimen beneath a spherical indenter can be first expected to occur when pm » 1.1Y. As load is further applied, the plastic zone grows in size and this results in a levelling off of the indentation stress. This indicates a departure from the linear elastic response.

Evolution of a fully developed plastic zone.

Figure 3. Evolution of a fully developed plastic zone.

Development of A Plastic Zone

There becomes a point where any increase in load results in a proportional increase in the contact radius and so the mean contact pressure becomes constant (ignoring any strain hardening effects). Under these conditions of a fully developed plastic zone, we call the mean contact pressure the hardness H of the specimen.

Transition between Elastic and Plastic Contact

The transition from elastic to plastic contact is of particular importance in relation to the testing of thin films. At very low penetration depths, an increase in hardness with increasing load is usually observed. This is due to the transition between elastic and plastic contact. In these initial stages of contact, the contact is elastic (or partially plastic) and the mean contact pressure rises with increasing a/R. Under these conditions, the quantity reported as the hardness H by a nanoindentation instrument should be more correctly called the mean contact pressure. Only when a fully developed plastic zone is obtained is the mean contact pressure equal to the hardness. This can usually be identified by a plateau in the values of H with increasing penetration depth. If the indenter is blunt, and the film is thin, then the substrate may influence the contact before a fully developed plastic zone can be achieved.

The Significance Developing The Plastic Zone

Understanding the significance of the developing plastic zone is critical to the use and interpretation of nanoindentation test data. Very often, users are not aware of this issue and ascribe changes in modulus or hardness to size effects, oxide layers and so on which are not present at all. It is the relationship between the indentation stress (the contact pressure) and the indentation strain that is the key issue.

Much more valuable information about nanoindentation can be found in Fischer-Cripps' free downloadable IBIS Handbook of Nanoindentation

This information has been sourced, reviewed and adapted from materials provided by Fischer-Cripps Laboratories.

For more information on this source, please visit Fischer-Cripps Laboratories.

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