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Elastic modulus vs Hardness plot

Another way to visualize the distribution of mechanical property results is to plot for example the elastic modulus (E) values vs the hardness (H) values. Such a plot leads sometimes to the observation of families of points and the definition of "sectors" or "bubbles", each one corresponding to a single phase (e.g. soft matrix vs hard and stiff particles).

The correlation between elastic and plastic properties has been extensively studied in the literature [5], [2], [11] and [7].


Elastic modulus is an intrinsic material property and hardness is an engineering property, which can be related to yield strength for some materials.

E-H map sectorization

As a first analysis of such a plot, sectors can be defined by giving an average value of elastic modulus and an average of hardness value, separating respectively by an horizontal line and a vertical line the different bubbles of points. Each sector is defined by a unique color.

Finally, average values of mechanical properties are given for each sectors directly into the graph, and a 4 color-coded map corresponding to this plot can be generated (see 2nd figure).


Example of sectorized elastic modulus vs hardness plot


Sectorized elastic modulus vs hardness plot with mean values and corresponding mechanical map

Clustering with Gaussian Mixture Models

The Gaussian mixture Models (GMM) are often used for data clustering [12]. This method is well described in the |matlab| documentation [8], [9] and [10] but also in the literature [4].

This method is powerful to separate contribution of 2 or 3 phases (especially in the case of a soft metallic matrix with hard ceramic particles) in the cloud of experimental points [6]. Average mechanical property values can also be extracted using this method and a 2 or 3 color map can be obtained too.

The |matlab| third party code used to define clusters GMM is: GMMClustering.m


Elastic modulus vs hardness plot with clusters of points obtained with GMM

Ashby map

Such plot could be seen as a conventional Ashby map [1], with materials families. An example of a typical Ashby map is given afterwards, using the CES Selector 2018 software [3]. At some point, it is possible to add material reference (bulk, homogeneous, monophasic, ...) values on the E-H map, in order to compare experimental data with data from the literature.


Typical Ashby map of elastic modulus vs Vickers hardness, obtained using CES Selector software


[1]Ashby M.F., "Materials Selection in Mechanical Design" (2005), ISBN 978-0-7506-6168-3.
[2]Bao Y.W. et al., "Investigation of the relationship between elastic modulus and hardness based on depth-sensing indentation measurements" (2004).
[3]CES Selector 2018
[4]Fraley C. and Raftery A.E., "How Many Clusters? Which Clustering Method? Answers Via Model-Based Cluster Analysis" (1998).
[5]Gent A.N., "On the Relation between Indentation Hardness and Young's Modulus." (1958).
[6]Hu C., "Nanoindentation as a tool to measure and map mechanical properties of hardened cement pastes" (2005).
[7]Labonte D. et al., "On the relationship between indenation hardness and modulus, and the damage resistance of biological materials" (2017).
[8]Mathworks - Gaussian Mixture Models
[9]Mathworks - Cluster
[10]Mathworks - Cluster Using Gaussian Mixture Models
[11]Oyen M.L., "Nanoindentation hardness of mineralized tissues" (2006).
[12]Wilson W. et al., "Automated coupling of NanoIndentation and Quantitative EnergyDispersive Spectroscopy (NI-QEDS): A comprehensive method to disclose the micro-chemo-mechanical properties of cement pastes" (2018).
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