Elsevier

Intermetallics

Volume 19, Issue 9, September 2011, Pages 1275-1281
Intermetallics

Modeling hardness of polycrystalline materials and bulk metallic glasses

https://doi.org/10.1016/j.intermet.2011.03.026Get rights and content

Abstract

Though extensively studied, hardness, defined as the resistance of a material to deformation, still remains a challenging issue for a formal theoretical description due to its inherent mechanical complexity. The widely applied Teter’s empirical correlation between hardness and shear modulus has been considered to be not always valid for a large variety of materials. The main reason is that shear modulus only responses to elastic deformation whereas the hardness links both elastic and permanent plastic properties. We found that the intrinsic correlation between hardness and elasticity of materials correctly predicts Vickers hardness for a wide variety of crystalline materials as well as bulk metallic glasses (BMGs). Our results suggest that, if a material is intrinsically brittle (such as BMGs that fail in the elastic regime), its Vickers hardness linearly correlates with the shear modulus (Hv = 0.151G). This correlation also provides a robust theoretical evidence on the famous empirical correlation observed by Teter in 1998. On the other hand, our results demonstrate that the hardness of polycrystalline materials can be correlated with the product of the squared Pugh’s modulus ratio and the shear modulus (Hv=2(k2G)0.5853 where k = G/B is Pugh’s modulus ratio). Our work combines those aspects that were previously argued strongly, and, most importantly, is capable to correctly predict the hardness of all hard compounds known included in several pervious models.

Graphical abstract

One sentence for highlighted figure: Vickers hardness has been theoretically evidenced to correlate successfully with shear modulus for various bulk metallic glasses (BMGs, left panel) and with a product of the squared Pugh’s modulus ratio and shear modulus for a wide variety of polycrystalline materials (including all superhard materials known, right panel).

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Highlights

► This work derived a theoretical formula of Vickers hardness linearly correlated with shear modulus for various bulk metallic glasses. ► This work derived a theoretical formula to predict Vickers hardness of a wide variety of polycrystalline materials. ► This work generalized the hardness formula through a thorough comparison for BMGs and polycrystalline materials. ► This work validated the powerful prediction of the proposed hardness formula. ► This work highlighted a comparison of several semi-empirical hardness models with the currently proposed formula.

Introduction

Despite the great efforts, to understand the theory of hardness and to design new ultrahard materials are still very challenging for materials scientists [1], [2], [3], [4]. During the past few years, several semi-empirical theoretical models [5], [6], [7], [8], [9] have been developed to estimate hardness of materials based on: (i) the bond length, charge density, and ionicity [5], (ii) the strength of the chemical bonds [6], (iii) the thermodynamical concept of energy density per chemical bonding [7], and (iv) the connection between the bond electron-holding energy and hardness through electronegativity [8], and (v) the temperature-dependent constraint theory for hardness of multicomponent bulk metallic glasses (BMGs) [9]. Experimentally, hardness is a highly complex property since the applied stress may be dependent on the crystallographic orientations, the loading forces and the size of the indenters. In addition, hardness is also characterized by the ability to resist to both elastic and irreversible plastic deformations and can be affected significantly by defects (i.e., dislocations) and grain sizes [10]. Therefore, hardness is not a quantity that can be easily determined in a well-defined absolute scale [1]. It has been often argued [13] that hardness measurements unavoidably suffer from an error of about 10%. All these aspects add huge complexity to a formal theoretical definition of hardness [5], [6], [7], [8], [9].

Within this context, to find a simple way to estimate hardness of real materials is highly desirable. Unlike hardness, the elastic properties of materials can be measured and calculated in a highly accurate manner. Therefore, it has been historically natural to seek a correlation between hardness and elasticity. The early linear correlation between the hardness and bulk modulus (B) for several covalent crystals (diamond, Si, Ge, GaSb, InSb) was successfully established by Gilman and Cohen since 1950s [10], [11]. Nevertheless, successive studies demonstrated that an uniformed linear correlation between hardness and bulk modulus does not really hold for a wide variety of materials [1], [12], [13], as illustrated in Fig. 1(a). Subsequently, Teter [12] established a better linear correlation between hardness and shear modulus (G), as illustrated in Fig. 1(b). This correlation suggests that the shear modulus, the resistance to reversible deformation under shear strain, can correctly provide an assessment of hardness for some materials. However, this correlation is not always successful, as discussed in Refs. [5], [13], [14]. For instance, tungsten carbide (WC) has a very large bulk modulus (439 GPa) and shear modulus (282 GPa) but its hardness is only 30 GPa [15], clearly violating the Teter’s linear correlation [see Fig. 1(b)] [5]. Although the link between hardness and elastic shear modulus can be arguable, it is certain to say that the Teter’s correlation grasped the key.

In this manuscript, following the spirit of Teter’s empirical correlation, we successfully established a theoretical model on the hardness of materials through the introduction of the classic Pugh modulus ratio of G/B proposed in 1954 [16]. We found that the intrinsic correlation between hardness and elasticity of materials correctly predicts Vickers hardness for a wide variety of crystalline materials as well as BMGs. Our results suggest that, if a material is intrinsically brittle (such as BMGs that fail in the elastic regime), its Vickers hardness linearly correlates with the shear modulus (Hv = 0.151G). This correlation also provides a robust theoretical evidence for the famous empirical correlation observed by Teter in 1998. On the other hand, our results demonstrate that the hardness of crystalline materials can be correlated with the product of the squared Pugh’s modulus ratio and the shear modulus (Hv=2(k2G)0.5853 where k is Pugh’s modulus ratio). This formula provides the firm evidence that the hardness not only correlates with shear modulus as observed by Teter, but also with bulk modulus as observed by Gilman et al. Our work combines those aspects that were previously argued strongly, and, most importantly, is capable to correctly predict the hardness of all compounds included in Teter’s [12], Gilman’s [4], [10], Gao et al.’s [5] and Šimůnek and Vackář’s [6] sets. Also, our model clearly demonstrates that the hardness of bulk metallic glasses is intrinsically based on the same fundamental theory as the crystalline materials. We believe that our relation represents a step forward for the understanding and predictability of hardness.

Section snippets

Model and results

According to Vicker [10], the hardness of Hv is the ratio between the load force applied to the indenter, F, and the indentation surface area:Hv=2Fsin(θ/2)d2,where d and θ are the mean indentation diagonal and angle between opposite faces of the diamond squared pyramid indenter, respectively (Fig. 2). In order to derive our model, we first assume that (i) the diamond squared pyramid indenter can be divided into four triangular based pyramid indenters and that (ii) the Vickers hardness is

Discussion and remarks

The hardness of a material is the intrinsic resistance to deformation when a force is applied [1]. Currently, a formal theoretical definition of hardness is still a challenge for materials scientists. The need for alternative superhard and ultrahard materials for modern technology has brought a surge of interest on modeling and predicting the hardness of real materials. In particular, in recent years several different semi-empirical models for hardness of polycrystalline covalent and ionic

Acknowledgements

We greatly appreciate useful discussions with Profs. Zhang Zhefeng, Jian Xu and Luo Xinghong in the IMR. X.-Q.C. acknowledges the support from the “Hundred Talents Project” of CAS and the NSFC (Grant No. 51074151). The authors also acknowledge the computational resources from the Beijing Supercomputing Center (including its Shenyang Branch in the IMR) of Chinese Academy of Sciences and the local HPC cluster of the Materials Process Modeling Division in the IMR.

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