Hardness Numbers and Stress-Strain Curves

Published on 24 July 2020

Hardness Testing

Hardness testing involves pushing a hard indenter into a sample with a known force and measuring the size of the residual indent.  Different types of hardness test differ only in the shape and size of the indenter.  For a given indenter and force, the indent size depends on the resistance of the sample to plastic deformation.  A hardness number is defined as the force divided by the area of contact between indenter and specimen.  A typical hardness test is that of Vickers, the geometry of which is shown in Fig.1.  The (diamond) indenter is a right pyramid with a square base and an angle of 136˚ between opposite faces.  

Hardness Vickers Plastometrex PIP
Fig.1 Geometry of the Vickers hardness test

The hardness number is the ratio of the force, F (in kgf) to the contact area (NOT the projected area), in mm2.  The indent diameter, D, taken as the average of D1 and D2, is measured in projection.  The value of HV is therefore given by

hardness vicker's
hardness vicker's

A simple calculation thus allows the hardness number to be obtained from the measured value of D.

Stress and Strain Fields during Indentation

The stress (in MPa) acting on the contact area is obtained on multiplying the force in kg by 9.81.  However, there is no simple relationship between this stress and the stress field in the material beneath the indenter, which depends in a complex way on indenter shape and plasticity characteristics of the material.  In fact, even for a given indenter shape, the hardness number varies with the applied force, which affects the level of plastic strain (dictating the parts of the stress-strain curve that affect the outcome).

Nevertheless, it is in practice quite common to use simple expressions to obtain a yield stress value from a hardness number.  This is based on the fact that, in the absence of work hardening, and for a given indenter shape, there is an approximately linear relationship between the yield stress (acting throughout the plastically deformed volume) and the hardness number.  For the Vickers test

hardness equation

Since very few materials exhibit no work hardening at all, a yield stress obtained in this way is not reliable.  In fact, it is likely to be indicative of a flow stress averaged in some way over the range of plastic strain experienced during the test, which depends on the depth of penetration, the indenter shape and the work hardening characteristics themselves.  It should never be regarded as more than a semi-quantitative indication of the resistance that the material offers to plastic deformation.

plastic strains indentation
Fig.2 Plastic strain fields after spherical indentation to 40% of the radius

Obtaining Stress-Strain Curves from Indentation

The core of Indentation Plastometry lies in detailed examination of the actual stress and strain fields around an indent.  This is illustrated by Fig.2.  Two materials like this might have rather similar hardness numbers (flow stresses averaged over an appropriate strain range  –  see Fig.3).  However, it can be seen that, while they might exhibit similar indent diameters, the residual indent shapes are very different (with much more “pile-up” for the sample exhibiting a low work hardening rate).  Repeated simulation of the indentation process allows convergence on the stress-strain curve (represented by a constitutive law) giving the best agreement with the indent profile.  Inferred stress-strain curves such as those below illustrate the dangers of relying on a hardness number.

sig eps schems
Fig.3 Stress-strain curves for two materials that might yield a similar HV (~ 3 times the flow stress averaged over a strain range of a few tens of %)

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