Glass hardness: Predicting composition and load effects via symbolic reasoning-informed machine learning

Sajid Mannan, Mohd Zaki, Suresh Bishnoi, Daniel R. Cassar, Jeanini Jiusti, Julio Cesar Ferreira Faria, Johan Frederik Schou Christensen, Nitya Nand Gosvami, Morten Mattrup Smedskjær, Edgar D. Zanotto*, N. M. Anoop Krishnan*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

5 Citations (Scopus)


Glass hardness varies in a non-linear fashion with the chemical composition and applied load, a phenomenon known as the indentation size effect (ISE), which is challenging to predict quantitatively. Here, using a curated dataset of over 3,000 inorganic glasses from the literature comprising the composition, indentation load, and hardness, we develop machine learning (ML) models to predict the composition and load dependence of Vickers hardness. Interestingly, when tested on new glass compositions unseen during the training, the standard data-driven ML model failed to capture the ISE. To address this gap, we combined an empirical expression (Bernhardt's equation) to describe the ISE with ML to develop a framework that incorporates the symbolic equation representing the domain reasoning in ML, namely Symbolic Reasoning-Informed ML Procedure (SRIMP). We show that the resulting SRIMP outperforms the data-driven ML model in predicting the ISE. Finally, we interpret the SRIMP model to understand the contribution of the glass network formers and modifiers toward composition and load-dependent (ISE) and load-independent hardness. The deconvolution of the hardness into load-dependent and load-independent terms paves the way toward a holistic understanding of the composition effect and ISE in glasses, enabling efficient and accelerated discovery of new glass compositions with targeted hardness.
Original languageEnglish
Article number119046
JournalActa Materialia
Publication statusPublished - 15 Aug 2023


  • Glass composition
  • Hardness
  • Indentation load
  • Machine Learning (ML), Indentation Size Effect (ISE)


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