Establishing a data-driven strength model for $\beta$-tin by performing symbolic regression using genetic programming

Highlights

• Established accurate data-driven strength models for tin that exhibit significant temperature and strain-rate dependent behavior by using genetic programming to perform symbolic regression.

• The developed model is robust and able to accurately predict the values of stress for wide range of strain rates, plastic strains and temperatures.

• Symbolic regression with genetic programming allowed for the development of useful strength models that are not limited by the constraints that traditional strength models have.

Abstract

Tin (Sn) exhibits complex deformation behavior characterized by significant dependence of strength on temperature and strain rate. This work develops a strength model for tin by using genetic programming to perform symbolic regression on a set of compression tests at various strain rates and temperatures. The strength model developed in this work showed increased accuracy compared to traditional strength models. Furthermore, the developed strength model adequately predicted independent experimental data (i.e., data that was not used to train the model). Results demonstrate that genetic programming successfully established a valid analytical function that adequately characterizes the temperature and strain rate dependent strength behavior of tin. Therefore, demonstrating that the developed framework provides robust and accurate formulations of strength models.

Introduction

Numerous strength models have been developed over the years to accurately describe the plastic constitutive behavior of metals [1], [2], [3], [4], [5], [6], [7]. The aforementioned models describe the material’s strength with different empirical mathematical formulations ranging from simple power and exponential laws [1], [2], to more complex formulas to incorporate test conditions (e.g. temperature, strain rate and pressure) [7], [8] and microstructural information (e.g. grain morphology and defects). As a result, traditional strength models leverage a-priori knowledge as well as domain expertise to establish a formulation that is general enough to describe the typical stress–strain curves of polycrystalline metal alloys. As a result, one typically needs to evaluate a large number of models when trying to characterize the behavior of a material. However, traditional strength models often fail to account for the specific factors that influence constitutive behaviors in certain materials given the general nature of the formulations [9]. Consequently, despite the many available options an accurate characterization of the behavior of the material of interest might not be possible since the available models do not explain accurately the behavior.

One could increase the complexity of the general formulation [8], [9], [10], [11]. However, this ad-hoc approach is often detrimental since increasing the complexity of the model increases the number of fitting parameters, which in turn requires additional experimental data to avoid over-fitting. Therefore, a critical need exists for a data-driven approach for establishing accurate strength models, especially for materials that show complex constitutive behavior such as tin.

Machine learning represents a viable solution to the above mentioned challenges given its unparalleled ability for learning complex and non-linear functional mappings that link the inputs to the desired outputs [12], [13]. Of specific interest to this work is genetic programming (GP), which is an extremely flexible evolutionary algorithm [14] capable of identifying optimal mathematical expressions that best conform to the observed data [15] by enabling one to perform symbolic regression [16], [17], [18], [19]. GP has been successfully implemented in different areas of material science [20], [21], [22], [23]. Furthermore, recent works have demonstrated the validity of GP to perform symbolic regression to characterize the plastic deformation of materials [24], [25], [26], [27], [28], [29], [30]. For example, Weber et al. [25] relied on symbolic regression to obtain an analytical expression for the coefficients of a crystal plasticity model. Recent studies by [27], [30] defined novel fitness metrics that enabled performing symbolic regression to find not only stress–strain curves but constitutive equations for plasticity models as well. Finally, there are studies by [24], [28], [29] where GP was used to perform symbolic regression and develop strength models from experimental data of different metals performed at different strain rates and temperatures. It is important to point out that the behavior captured by the aforementioned models successfully reproduced typical stress–strain response, i.e., monotonic increase and saturation of the strength. Nevertheless, the modularity and efficacy of the GP make it an ideal tool to obtain a strength model that best conforms to the data regardless of the complexity of the behavior observed.

In this work, we build upon the previous successes to develop a protocol that performs symbolic regression to obtain accurate and robust strength models for materials that traditional strength models struggle to characterize accurately. Specifically, we demonstrate the developed protocol on the complex stress–strain behavior of tin that has significant temperature and rate dependent strength as well as a drastic change in hardening behavior from low strain-rates to high strain-rates [31]. This works enables the development of empirical strength models in a fully data-driven and automated manner by leveraging GP. The developed protocol performs a thorough exploration of the parameters that guide the GP-based symbolic regression and helps identify the optimal model for the observed experimental data.

The data-driven model established in this work was trained on existing stress–strain data of tin at various temperatures and strain rates [31]. The accuracy and robustness of the established data-driven model was then critically validated by comparing its predictions to experimental results on which the model was not trained. Lastly, the accuracy of the developed data-driven model was compared to two traditional strength models (Johnson–Cook [32], [33] and Zerilli–Armstrong [34]), calibrated to the same experimental data.