Elastic modulus vs Hardness plot ================================= .. include:: includes.rst 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 [#Gent_1958]_, [#Bao_2004]_, [#Oyen_2006]_ and [#Labonte_2017]_. .. note:: 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). .. figure:: ./_pictures/MTS_example1_25x25_H_GUI_12.png :scale: 50 % :align: center *Example of sectorized elastic modulus vs hardness plot* .. figure:: ./_pictures/sectorMap.png :scale: 50 % :align: center *Sectorized elastic modulus vs hardness plot with mean values and corresponding mechanical map* Automated cluster analysis (K-means, Gaussian Mixture, ...) ############################################ Cluster analysis (or clustering) is an unsupervised machine learning task. For example, K-Means and Gaussian Mixtures (GMs) are both popular cluster analysis model for nanoindentation datasets. K-Means models +++++++++++++++++++++++++++ The K-Means are often used for nanoindentation data clustering [#Koumoulos_2019]_, [#Konstantopoulos_2020]_, [#Alhamdani_2022]_ and [#Jentner_2023]_. Like explains in [#Koumoulos_2019]_, "the K-Means algorithm aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. The number of clusters is known and each point can belong only in one cluster. At first, random k cluster center points are generated and each data point is assigned to the cluster with the nearest center point (smallest Euclidean distance). Then, the mean of each cluster is calculated and the k cluster centers are replaced by the corresponding cluster mean. Again, each point is assigned to the nearest cluster, measured in Euclidean distance." This method is as well described in the |matlab| documentation [#Matlab_KM]_. A possible |matlab| third party code, which could be used to define clusters with K-Means model is: https://www.mathworks.com/matlabcentral/fileexchange/24616-kmeans-clustering?s_tid=mwa_osa_a Gaussian mixture models +++++++++++++++++++++++++++ The GMs model are as well used for nanoindentation data clustering [#Wilson_2018]_ and [#Chen_2021]_. This method is well described in the |matlab| documentation [#Matlab_GMM]_, [#Matlab_cluster]_ and [#Matlab_clustering]_ but also in the literature [#Fraley_1998]_. 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 [#Hu_2005]_. Average mechanical property values can also be extracted using this method and a 2 or 3 color map can be obtained too. The influence of indentation size and spacing on statistical phase analysis has also been studied by fast mapping indentation anc clustering analysis [#Besharatloo_2021]_. The |matlab| third party code used to define clusters with GMs model is: `GMMClustering.m `_ .. figure:: ./_pictures/clusterMap.png :scale: 50 % :align: center *Elastic modulus vs hardness plot with clusters of points obtained with GMM* Determination of the number of clusters +++++++++++++++++++++++++++ The optimal number of clusters (k) can be determined using the elbow method. This method looks at the total within-cluster sum of square (WSS) as a function of the number of clusters. One should choose a number of clusters so that adding another cluster doesn't improve much better the total WSS. This method can be defined as follow in 4 steps: 1) Compute clustering algorithm (e.g., K-Means or GMs algorithms) for different values of k. 2) For instance, by varying k from 1 to 5 clusters. For each k, calculate the total WSS. 3) Then, plot the curve of WSS according to the number of clusters k. 4) Finally, the location of a bend (knee) in the plot is generally considered as an indicator of the appropriate number of clusters. Note that, the elbow method is sometimes ambiguous. Alternatives are for example the average silhouette method or the gap statistic method... Next steps: Ashby map or self-organized maps ############################################ The next step after the different analytical, sectorization, clustering... approaches, could be to use nanoindentation outputs (phase mechanical properties in the case of a composite, an alloy...) for material selection, material design, material discovery.... A common strategy for material selection is the usage of conventional Ashby map [#Ashby_2005]_. An example of a typical Ashby map is given afterwards with materials families (envelops around different materials), using the CES Selector 2018 software [#CES_Selector]_. 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. .. figure:: ./_pictures/E-H_Ashby.png :scale: 50 % :align: center *Typical Ashby map of elastic modulus vs Vickers hardness, obtained using CES Selector software* Regarding material design or material discovery, an option is the usage of self-organized maps (SOMs) [#Qian_2019]_ in the framework of material informatics. Here an example for Atomic Force Microscopy (AFM) technique [#Weber_2023]_. References ############################################ .. [#Alhamdani_2022] Alhamdani S.G. et al., "Cluster-Based Colormap of Nanoindentation Using Machine Learning" (2022), IEOM Society International. .. [#Ashby_2005] Ashby M.F., "Materials Selection in Mechanical Design" (2005), ISBN 978-0-7506-6168-3. .. [#Bao_2004] `Bao Y.W. et al., "Investigation of the relationship between elastic modulus and hardness based on depth-sensing indentation measurements" (2004). `_ .. [#Besharatloo_2021] `Besharatloo H. and Wheeler J.M., "Influence of indentation size and spacing on statistical phase analysis via high‑speed nanoindentation mapping of metal alloys" (2021). `_ .. [#CES_Selector] `CES Selector 2018 `_ .. [#Chen_2021] `Chen X. et al., "Clustering analysis of grid nanoindentation data for cementitious materials (2021). `_ .. [#Fraley_1998] `Fraley C. and Raftery A.E., "How Many Clusters? Which Clustering Method? Answers Via Model-Based Cluster Analysis" (1998). `_ .. [#Gent_1958] `Gent A.N., "On the Relation between Indentation Hardness and Young's Modulus." (1958). `_ .. [#Hu_2005] `Hu C., "Nanoindentation as a tool to measure and map mechanical properties of hardened cement pastes" (2005). `_ .. [#Jentner_2023] `Jentner R.M. et al., "Unsupervised clustering of nanoindentation data for microstructural reconstruction: Challenges in phase discrimination" (2023). `_ .. [#Konstantopoulos_2020] `Konstantopoulos G. et al., "Classification of mechanism of reinforcement in the fiber-matrix interface: Application of Machine Learning on nanoindentation data" (2020). `_ .. [#Koumoulos_2019] `Koumoulous E.P. et al., "Constituents Phase Reconstruction through Applied Machine Learning in Nanoindentation Mapping Data of Mortar Surface" (2019). `_ .. [#Labonte_2017] `Labonte D. et al., "On the relationship between indenation hardness and modulus, and the damage resistance of biological materials" (2017). `_ .. [#Matlab_GMM] `Mathworks - Gaussian Mixture Models `_ .. [#Matlab_KM] `Mathworks - K-Means Models `_ .. [#Matlab_cluster] `Mathworks - Cluster `_ .. [#Matlab_clustering] `Mathworks - Cluster Using Gaussian Mixture Models `_ .. [#Oyen_2006] `Oyen M.L., "Nanoindentation hardness of mineralized tissues" (2006). `_ .. [#Qian_2019] `Qian J., "Introducing self-organized maps (SOM) as a visualization tool for materials research and education" (2019). `_ .. [#Weber_2023] `Weber A., "Application of self-organizing maps to AFM-based viscoelastic characterization of breast cancer cell mechanics" (2023). `_ .. [#Wilson_2018] `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). `_