Atomic mixed-mode cohesive-zone dual constitutive laws of impurity-embrittled grain boundaries in polycrystalline solids via nanoscale field projection method

Highlights

• Cohesive-zone dual constitutive laws of grain boundaries in nickel are obtained.

• Embrittlement of grain boundaries by sulfur impurity atoms is investigated.

• Atomic-scale features contribute significantly to the material fracture behaviors.

• New functional relationships for cohesive-zone dual constitutive laws are proposed.

Abstract

Atomic-scale mixed-mode intergranular fracture, featured by non-local non-linear discrete atomic debonding processes near a crack tip along a grain boundary (GB), is modeled with a cohesive zone in a continuum scale controlled by cohesive-zone dual constitutive relations, a balanced traction-separation relationship, i.e., a conventional cohesive-zone law (CZL), and an unbalanced traction (UT)-centerline displacement (CD) relationship. In order to bridge two different scales, we developed a nanoscale field projection method (nano-FPM) based on atomic-scale interaction J and M integrals, for which asymptotic anisotropic elastic fields near an interfacial crack with balanced and unbalanced crack-face tractions are used as probing fields of the CZL and UT-CD relationship, respectively. Cracking phenomena along a GB in nickel with segregated sulfur impurity atoms under mixed-mode loadings are simulated with molecular dynamics. Embrittlement by sulfur impurity atoms is quantitatively estimated with the mixed-mode CZL of the GBs in nickel via the nano-FPM developed in this study. UT-CD relationship, a special feature of atomic fracture, representing the micromechanical change of surface stress between GB and cracked surfaces, is also obtained by the nano-FPM. New functional relationships for CZL, UT-CD relationship, and decohesion potential obtained by the nano-FPM are proposed to facilitate the implementation of them into mesoscale or continuum-scale analysis on mixed-mode intergranular fracture.

Introduction

Impurity segregation to grain boundaries (GBs) driven by thermodynamic potential energy may change the microstructure and local chemical bonds of polycrystalline solids, resulting in dramatic changes in their mechanical properties, for instance, the strength and fracture toughness (Buban et al., 2006; Chen et al., 2010; Duscher et al., 2004; Gianola et al., 2008; Shi et al., 2017; Tehranchi and Curtin, 2017; Všianská and Šob, 2011). Segregated impurities, e.g., substitutional Al and interstitial Si and P into GBs in Ni, may enhance the cohesion and thus fracture toughness of Ni GBs (Všianská and Šob, 2011), since the presence of impurities increases the number of bonds and bond strength along the GBs (Buban et al., 2006). On the other hand, some segregated impurities may lead to the GB embrittlement and facilitate the intergranular fracture of polycrystalline solids (Heuer et al., 2002; Rogers, 1968; Schweinfest et al., 2004; Yamaguchi et al., 2005). Numerous studies have been conducted to determine the effect of impurity segregation on the mechanical behavior of GBs (some examples are given in (Barrows et al., 2016; Duscher et al., 2004; Kuhr et al., 2016; Laporte and Mortensen, 2009; Ludwig et al., 2005; Petucci et al., 2014; Razumovskiy et al., 2015; Rice and Wang, 1989; Siegel and Hamilton, 2005; Tehranchi and Curtin, 2017)). For example, the impurity atoms such as S and H segregated into GBs in Ni can reduce the ductility and cause ductile-to-brittle transition behavior (Heuer et al., 2002; Lassila and Birnbaum, 1986, 1987). The experiments by Heuer et al. (2002) indicated that the critical sulfur concentration of 15.5 ± 3.4% exists for intergranular embrittlement in nickel, and a similar critical concentration of 14.2 ± 3.3% exists for impurity-induced amorphization. Although experimental studies clearly demonstrate embrittlement by segregated impurities, they do not provide microstructural information and evolution at the atomic scale. Later, atomistic simulations including first principles methods have been increasingly used to provide additional insights into the microstructural change and embrittlement of GBs induced by impurity segregation (Barrows et al., 2016; Kuhr et al., 2016; Schweinfest et al., 2004; Siegel and Hamilton, 2005; Tehranchi and Curtin, 2017; Yamaguchi et al., 2005; Yuan et al., 2011). Their results showed that the impurity-induced changes in electronic structure lead to the weakening of the bonds holding the GBs, and the impurity embrittlement effects can be estimated in terms of the reduction of work of separation, which is the energy difference of impurities at a GB and at a free surface according to the Rice-Wang model (Rice and Wang, 1989). Another embrittlement mechanism related to electronic-structure changes is the stiffening of the GBs, by strengthening the directional bonds and hindering the mobility of bonds and dislocations (Goodwin et al., 1988; Haydock, 1981). In many cases, however, the impurity-induced embrittlement can be caused by the size effect, i.e., GB expansion that leads to the weakening of the GBs (Lozovoi et al., 2006; Schweinfest et al., 2004).

In general, atoms at and near a free surface of a solid experience a different local electron density than those in the bulk due to the loss of their neighbors. Therefore, these surface atoms tend to increase the surrounding charge density by changing the interatomic distance and creating a surface stress. The concepts and physics of surface stress are firstly introduced by Gibbs (1906), in which the surface stress relates the reversible work per unit area needed to elastically stretch a pre-existing surface, or associates with the variation of the excessive free energy to the variation of the surface strain (Cammarata, 1994, 1997). In order to evaluate the surface stress of clean surface at atomic scale, the surface strain dependence of the excessive free energy for different metals can be determined based on first-principle calculations (Fiorentini et al., 1993; Ibach, 1997; Needs et al., 1991) and semiempirical calculations (Ackland and Finnis, 1986; Ibach, 1997; Stepanyuk et al., 2000). It is worth noting that the surface stress can also be affected by the impurity adsorption since the interaction between an adsorbate and the surface may induce a structural rearrangement (Haiss, 2001; Stepanyuk et al., 2000). Based on the importance of surface phenomena, a rational theory of continuum mechanics for material surfaces has been developed by Gurtin and Ian Murdoch (1975), in which the surface region is modelled as a membrane-type surface; that is, a material surface has no resistance to bending. Then, the continuum theory of material surfaces has been generalized by Steigmann and Ogden (1999) by including the effect of flexural resistance. At a smaller scale, the effect of surface stresses has recently received much attention in research on the mechanical responses of nanostructured materials, e.g., thin films, multilayers, nanoparticles and nanocrystals (Fischer et al., 2008; Levitas et al., 2018; Liang et al., 2018; Stojić and Binggeli, 2012; Style et al., 2017; Xu et al., 2017; Zadin et al., 2018). Recently, the effect of surface stress in solids containing cracks has been considered (Choi and Kim, 2007; Fu et al., 2008, 2010; Kim et al., 2009; Wang et al., 2007; Wu, 1999; Wu and Wang, 2000), since the surface stress is believed to influence the stresses in the vicinity of the crack tip. For example, Wang et al. (2007) included both the surface stress and surface elasticity to solve the mode-I and mode-III crack problems analytically, and then examined the surface effect on the elastic field near the crack tip. Wu and Wang (2000) indicated that the surface stress may be considered as the applied point force at a crack tip. The placement of the point force tends to change the potential energy of the system, and alter the configurational equilibrium condition, resulting in a reduction of the fracture toughness of a solid. Choi and Kim (2007) pointed out that surface stress is responsible for the traction jump across the cracked surfaces in the nanoscale cohesive-zone model (CZM), and then quantitatively estimated the surface stress in the cohesive zone based on the stress balance at the crack interface.

Recent progress and interest in micro- and nano-mechanics of solids require details of the cohesive-zone law (CZL), i.e., the shape of traction-separation curves within cohesive zone, for studying cracking phenomena with high resolution. The CZL can be regarded as a phenomenological model instead of an exact physical representation of material behavior in the fracture process zone, including complex non-linear fracture phenomena in ductile materials or quasi-brittle materials (Barenblatt, 1962; Dugdale, 1960; Maugis, 1992). To investigate the effects of impurity embrittlement on intergranular fracture, several researchers have tried to extract the traction-separation relationships by the rigid separation potential method (Yamaguchi et al., 2005) or directly from the atomic stresses and displacements near the crack tip (Barrows et al., 2016). These direct methods may not provide proper CZL, since continuum mechanics concept is not valid in the highly non-linear non-local discrete atomic fracture process zone. Therefore, the high atomic stresses from the atomic fracture process zone may enter the cohesive tractions, and thus, it is challenging to use this CZL with unrealistic high traction in upper length scales, i.e., mesoscale and continuum scale. One promising method to extract the crack-tip CZL from the elastic far fields is proposed by Hong and Kim (2003), named as field projection method (FPM). This technique utilizes the interaction J integral between the measured fields and a series of auxiliary eigenfunctions for reconstructing the traction-separation relationships at the crack tip in a homogenous isotropic solid. Subsequent studies (Chew et al., 2009; Hong et al., 2009) demonstrated that the FPM can provide a systematic way to determine traction-separation curves of various micromechanical fracture processes. Later, the FPM was developed for non-linear elastic or elasto-plastic materials by using the Maxwell-Betti's reciprocal theorem together with the Fourier series (Chew, 2013). Choi and Kim (2007) have also extended the FPM to investigate the nanoscale fracture process in crystalline materials by using a series of eigenfunctions for an interface cohesive crack tip field in anisotropic materials. Since the deformation tends to develop in a low symmetry system at the nanoscale, the elastic fields surrounding a cohesive zone should be analyzed with anisotropic elasticity. One limitation of this FPM is the convergence issue due to the high-order eigenfunction representations of the traction and the separation-gradient distribution. Recently, Kim et al. (2012) have proposed a new set of auxiliary fields corresponding to the piecewise uniform tractions on each element face of the cohesive crack surfaces and applied interaction J and M integrals with the new auxiliary fields to probe the CZL of real fields. This approach solved the convergence issue, but the new set of auxiliary fields was obtained numerically but not analytically. However, the FPM in its current development is hard to apply for analyzing the fracture processes at the atomic level, and the contribution of the surface stress in the cohesive zone has never been properly considered. In atomic decohesion processes, the surface stress can sustain the unbalanced traction (UT), i.e., the traction jump across the cracked surfaces, and thus not only the conventional CZL but also the UT with its proper work-conjugate, i.e., the centerline displacement (CD) of cracked surfaces, should be included in cohesive-zone constitutive relations.

In intergranular fractures of brittle solids, cracks propagate mainly along GBs even under mixed-mode loadings (Alfaiate and Sluys, 2017; Aliha et al., 2017; Cui et al., 2017; Katanchi et al., 2018) so that intergranular fractures can generally be considered as mixed-mode fracture, and thus, the cohesive-zone constitutive relations of GBs are expected to depend on the mode mixity. In order to analyze the mixed-mode fracture processes, several coupled and uncoupled CZLs have been widely adopted in the literature (Kafkalidis and Thouless, 2002; Li et al., 2006; Tvergaard and Hutchinson, 1992; Xu and Needleman, 1993; Zhou et al., 2009, 2008). For example, Xu and Needleman (1993) defined a complex exponential form to couple the opening and shear modes that allows for different fracture energies in different mode mixities. Under mixed-mode loadings, the areas under the traction-separation curves of mode I and mode II can be used to identify the respective energy release rate (ERR), and the total ERR is the sum of ERR in mode I and II. Although the cohesive models have been extended to investigate the mixed-mode fracture processes, however, to date, the mode dependence of a conventional CZL and an UT-CD relationship for cracking phenomena along a GB with segregated impurity atoms has not been discussed in the literature.

In this study, we have developed the nano-FPM based on interaction J and M integrals to reconstruct the atomic-level mixed-mode CZL of brittle interfacial cracks. Dual functional parameters (a balanced traction-separation relationship, and an UT-CD relationship) are proposed as the cohesive-zone constitutive relations of atomic fracture processes. In this augmented nano-FPM, anisotropic elasticity solutions to the interfacial crack in anisotropic bimaterials with balanced and unbalanced crack-face tractions are used as the analytic auxiliary fields. As a model problem of atomic mixed-mode fractures, GB cracking phenomena in nickel with the presence of the sulfur impurities are simulated with Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) program (Plimpton, 1995), in which the reactive force field (ReaxFF) (Nielson et al., 2005) is used to describe the atomic interaction for nickel and sulfur impurity atoms. Additionally, the new functional relationships for CZL, UT-CD relationship, and decohesion potential obtained by the nano-FPM are proposed to facilitate the implementation of them into mesoscale or continuum-scale analysis on mixed-mode intergranular fracture.