The optimum grain size for strength-ductility combination in metals

Highlights

• An optimum grain size ($d_{optimum}$) on the order of a few micrometers exist ubiquitously for strength-ductility synergy, at which the strain energy density limit reaches a maximum while maintaining high yield strength.

• Theoretical models on the grain size-dependence of uniform elongation and ultimate strength are developed, which accurately predict the $d_{optimum}$.

• The $d_{optimum}\approx$2$l_{Gbar}$, where $l_{Gbar}\approx \frac{k_{HP}{R}^{3/2}}{\sqrt{2}MGb}$ is the characteristic width of grain boundary affected region.

• The $d_{optimum}$ is the critical grain size having the strongest intragranular plastic strain gradient effect.

• The $d_{optimum}$ is the limiting dimension that allows toughening by grain refinement.

Abstract

A strength-ductility trade-off usually occurs when grains are refined to increase strength. A question arises on if there exists a grain size for the best strength-ductility combination, i.e., with the highest possible strain energy density limit and strength simultaneously. This issue is crucial for guiding the design of strong and tough structural materials. Here we reveal an optimum grain size ($d_{optimum}$) on the order of a few micrometers, at which the strain energy density limit, estimated as the product of strength and uniform elongation, reaches a maximum while maintaining reasonably high yield strength. The $d_{optimum}$ is found to exist in a series of single-phase FCC, BCC and HCP materials, indicating it as a universal phenomenon. Theoretical models on the grain size-dependence of uniform elongation and ultimate strength are developed by considering dislocation accumulation in grain boundary affected region (Gbar) and grain interior based on the classical Kocks-Mecking-Estrin model. Combined with the Hall-Petch relationship, the models accurately predict the $d_{optimum}$. Importantly, the models disclose this $d_{optimum}$ to be close to twice of the characteristic width of Gbar ($l_{Gbar}$), suggesting that it is exactly at or near the critical grain size with the strongest intragranular strain gradient effects.

Introduction

Metallic materials are desired to be strong and ductile at the same time. Unfortunately, these two properties are usually mutually exclusive, especially when manipulating grain size ($d$) to improve one of them (Meyers et al., 2006). Good ductility originates from high work hardening capability, which helps to resist strain localization and consequently prevent catastrophic failure (Ritchie, 2011; Zhu and Wu, 2018). This is one of the key reasons why coarse-grained (CG) materials remain the primary choice for most structural applications. However, the low strength of CG material requires more materials to carry the load, which makes it less energy efficient in applications (Li and Lu, 2017). Grain refinement can effectively strengthen materials, but it usually sacrifices ductility. As extreme cases, ultrafine-grained (UFG) and nanostructured (NS) materials can be many times stronger than their CG counterparts, but their ductility is typically lower than 5% (Lu, 2014; Ovid'ko et al., 2018). This presents a major problem for the high-strength NS bulk to serve in safety-critical applications. Such grain size-related strength-ductility trade-off raises a few critical issues. Does an optimum grain size ($d_{optimum}$) exist for strength-ductility combination? If so, how to determine and predict the $d_{optimum}$ theoretically? Furthermore, what is the mechanism behind it?

These issues are not only fundamental to understanding the grain size effects on mechanical behavior, but also important for guiding the microstructural design of high-performance materials, including the conventional engineering materials and advanced composites. For example, in the heterostructured materials, superior strength-ductility combination is expected if the grain size of the heterogeneous constituent zones can be tuned to the corresponding $d_{optimum}$ (Sathiyamoorthi and Kim, 2020; Zhu et al., 2021). Despite the relentless efforts in optimizing strength-ductility combination by largely manipulating the microstructures (Li and Lu, 2017; Wu et al., 2022, 2015; Zhong et al., 2022), the chemistry (Li et al., 2016; L. Y. Liu et al., 2022), or both at the same time (Fan et al., 2022; S. S. Liu et al., 2022), the present issues remain unsettled in theoretical understanding.

Achieving the best strength-ductility combination generally means that at the highest possible strength level, the strain energy density limit reaches or approaches the maximum. The difficulty in predicting the $d_{optimum}$ lies in the complexity of grain size effects on the strength and ductility that govern the strain energy density limit. Although the yield strength can be simply evaluated using the Hall-Petch equation (Hansen, 2004; Meyers and Chawla, 2009), the dependence of ductility and ultimate strength on grain size are not well understood (Li and Cui, 2007). Since ductility and ultimate strength are deformation history-dependent properties governed by work hardening and plastic instability (Yasnikov et al., 2022; Zhu and Wu, 2018), their dependence on grain size is primarily due to the interaction between dislocations and grain boundary. At the plastic stage, grain boundaries act as extra barriers to block dislocation motion and sites for dynamic recovery, resulting in distinct dislocation accumulations in the vicinity of grain boundary and in the grain interior (Delincé et al., 2007; Haouala et al., 2018; Hirth, 1972; Meyersm and Ashworth, 1982). Along with these behaviors, significant intragranular strain inhomogeneity is developed, which complicates the distribution of both short- and long-range internal stresses within grains (Ashby, 1970; Jiang et al., 2022). Grain refinement embraces more extensive grain boundary-dislocation interactions. As extreme cases, when the grain size is reduced to ultrafine or nanometer scale, the free slip path and storage room of dislocations are largely limited by the high-density grain boundaries, and the dynamic recovery is gradually dominated by grain boundaries (Meyers et al., 2006; Yu et al., 2005). The coupling of these factors complicates the interpretation of grain size effects on work hardening, making it a great challenge to derive explicit models on the grain size dependence of ductility and ultimate strength, and thus obscuring the reasoning path towards $d_{optimum}$.

Nevertheless, some recent experimental results may provide a potential hint on the possible $d_{optimum}$. Specifically, fine grains (FG) of a few micrometers were coincidentally involved in a variety of advanced structures with superior strength-ductility combination, including in homogeneous materials (Guo et al., 2022; Li et al., 2008; M.W. Liu et al., 2022; M.S. Wang et al., 2022), heterostructured materials (Huang et al., 2018; Wang et al., 2002; Wu et al., 2015), and conventional composites (Wu et al., 2017). For instance, in partially recrystallized Ti and Cu, both of which unite the UFG strength and the CG ductility, the FG zones with a grain size of about 2–4 μm were believed to play the key role in retaining work hardening (Wang et al., 2002; Wu et al., 2015). In the recently designed strong-yet-ductile manganese steel and complex-concentrated Ni-Fe alloys, FGs were the main constituent (Fan et al., 2022; Zhong et al., 2022). There appears to be some kinds of efficient strengthening and hardening mechanisms in this size range. Unfortunately, most of these structures were designed by trial and error, lacking clear physical guidelines on the grain size. In short, the confirmation of the existence and the exact value of $d_{optimum}$ as well as the possible deformation physics behind it remain elusive.

In this work, experimental results on the tensile properties of a series of single-phase materials with various crystal structures are collected over a wide grain size range, from extensive literature available to the authors. Systematic analyses on grain size effects verify that there indeed exists universally a $d_{optimum}$ at which the optimal strength-ductility combination is achieved. Theoretical models on grain size effects are established to predict the $d_{optimum}$. Deformation physics potentially responsible for the superior properties at $d_{optimum}$ will be discussed. Note that the $d_{optimum}$ is currently discussed in terms of the quasi-static tension properties at ambient temperature, with dislocation-dominated plasticity, in order to deliver more general fundamentals.