1 Introduction

Understanding of crack propagation through grain boundaries (GBs) and interfaces in polycrystalline materials has been of substantial interest in the field of fracture mechanics. In a typical polycrystalline metal, GBs are significantly thinner (\({\sim }\)10 nm) than the grains (\({\sim }\upmu \hbox {m}\)). The accurate prediction of crack propagation through GBs and interfaces while simultaneously predicting crack propagation though grains can be difficult due to the factors that include: significant difference in the length scales of GBs and grains, respectively; unknown GB strength properties; embrittlement effects of GBs owing to GB chemistry etc. The latter occurs due to the chemical segregation of impurity elements (Luo et al. 2005; Tamaki et al. 1994). The impact of GBs on the fracture strength of binary \(\hbox {Li}_{2}\)-type alloys has been investigated earlier (Takasugi and Izumi 1985), using mechanical tests combined with metallographic and fractographic observations. This investigation facilitated development of the relationship between the structure and strength of GBs in Si bi-crystals (Sato et al. 1990), and combined the fracture strength measurements with transmission electron microscopy (TEM) observations. In the work of Kong and Qiao (2005), the effect of GBs on fracture resistance was measured experimentally using cleavage fracture tests. It was found that when a crack cleavage front encounters a high-angle GB, it initially penetrates through the GB at a number of breakthrough points. For high-angle GBs, persistent GB islands can act as bridging reinforcement. It has been reported that sintering of refractory metals such as W results in significant GB embrittlement (Gupta et al. 2007). In addition, the GB region is significantly harder relative to the grain interior in polycrystalline W (Westbrook and Wood 1961). The GB embrittlement occurs from the creation of a barrier to dislocation propagation by formation of a hardened region near GB (Lee and Tomar 2013, 2014, 2012). Considering the fact that Ni additions are likely to be segregated along GBs in polycrystalline W (Gupta et al. 2007), change in GB mechanical property plays a significant role in enhancing or degrading of the materials failure resistance. Although the effects of relative orientation of grains undoubtedly exist, they appear to be small, since comparable hardening is observed in all boundaries of randomly oriented polycrystalline samples. A usual requisite of all analyses discussed so far is a necessity to understand how the chemical structure of GBs correlates to the fracture strength of the material, e.g. Luo et al. (2005), Margevicius et al. (1999), Ray et al. (1994), Gumbsch (2003). The present work focuses on predicting effect of GB strength on crack propagation resistance in polycrystalline W.

A number of analytical criteria have been proposed to predict crack propagation. Linear elastic fracture mechanics has been used to calculate mixed-mode stress intensity factor (SIF) to predict incremental changes in trajectory, and to assess local crack stability (Bittencourt et al. 1996). A criterion for the direction of crack propagation normal to the direction of maximum hoop stress was proposed by Erdogan and Sih (1963). The strain energy and the energy release rate have been used to propose a new criteria in the works of Sih (1974) and Nuismer (1975), respectively. In the work of Benthem (1977), a new way of formulating analytical solution is introduced using the spherical coordinate system. Later, singular stress field near the terminal point of the crack front edge at the surface of an elastic body to predict crack propagation has been investigated by Bažant and Estenssoro (1979). In order to perform numerical prediction of crack propagation, the enriched finite element model (EFEM) has been introduced with emphasis on enriching the near crack-tip field using the asymptotic solution for static fracture problems (Benzley 1974). An assumed displacement hybrid finite element model has also been developed by Atluri et al. (1975). Furthermore, the EFEM has been extended to direct calculation of combined mode I and II SIFs for axisymmetric and planar structures of arbitrary geometry and loading in the work of Nash Gifford Jr and Hilton (1978). Later, discontinuous enrichment functions have been added to the FEM approximation in order to account for the presence of crack and allow the crack to be arbitrarily aligned within a FE mesh by Belytschko and Black (1999). This methodology was improved, making it possible to independently represent the entire crack from the mesh in the work of Moës et al. (1999) that also introduced Extended finite element method (XFEM). XFEM has been improved by various researchers to efficiently and realistically represent crack growth and discontinuities independent of element boundaries (Dolbow et al. 2000, 2001; Sukumar et al. 2000; Gravouil et al. 2002; Moës et al. 2002; Huang et al. 2003; Sukumar and Prévost 2003). For crack propagation modeling in XFEM, a discontinuous function and near-tip asymptotic functions are added to the FEM approximation using the framework of partition of unity. Although there are a number of available formulations to simulate crack propagation, XFEM can provide direct evaluation of mixed mode SIFs (Liu et al. 2004). Another benefit of XFEM is that, a very fine mesh is not required, particularly in the 3-D cases in order to obtain acceptable solutions in the vicinity of a singularity (Natarajan et al. 2010). The present work uses XFEM for crack propagation path prediction. Since the focus of the research is on crack path prediction and SIF as well as J-integral calculations, the issue of dislocation dynamics at the local crack tip and related crack initiation aspects are not considered.

The effect of GB embrittlement in polycrystalline W is often characterized in terms of brittleness index (BI). The BI is defined as the ratio between hardness and fracture toughness to have a certain generality in the formulation of the ductility verses brittleness question (Lawn and Marshall 1979; Evans and Wilshaw 1976). While the grain size of W ranges between 100 nm and 50 \(\upmu \)m (Wei et al. 2006), the GB thickness is known to have a range of 0.3–0.6 nm (Luo et al. 2005; Gupta et al. 2007). Such a big difference in the length scale of GBs and grains, respectively, makes a finite element-based analysis of a polycrystalline sample fracture while also accounting for the GB chemical structure very difficult. In the absence of such analyses, the BI has been utilized as a guiding parameter to predict the extent of brittleness of a given polycrystalline microstructure. The hardness is accounted as a measure of resistance to deformation, while the fracture toughness is defined as resistance to fracture. However, the relative scales of the deformation and fracture zones about a sharp-contact site introduce a size effect into the mechanical responses, which can also be interpreted as the ratio between the two different physical quantities (hardness and fracture toughness) and has a dimension of \(\upmu \hbox {m}^{-1/2}\). While the fracture toughness is a key value that significantly depends upon the brittleness of material, the measurement method of fracture toughness highly influences the test results owing to significant size and loading type dependencies. After the work of Irwin (1957) (SIF) and Rice (1968) (J-integral), experimental methods have been standardized by the American Society for Testing and Materials (ASTM) (ASTM-E1823 1996; ASTM-E399 1997). In a large number of cases, GB segregation occurs with a significant hardening of the GB region relative to that of the grain interior (Westbrook and Wood 1961). Such hardening of the GBs is directly correlated with the level of plasticity within the GB region, which may result in the GB embrittlement effect. A number of experimental studies have reported that both intergranular and transgranular fractures can be observed in polycrystalline W (Lee and Tomar 2012; Gumbsch 2003; Bartolucci and Schlössin 1966; Zurek and Gray III 1991). The impurity contents in W material Benthem (1977) is found to be one of the reasons for this inconsistency. Considering the fact that impurities within the GB region are not perfectly distributed within GBs, the influence of GBs on plastic deformation as well as intergranular vs. transgranular crack propagation varies in the same material (Krasko 1993). Another factor that influences GB embrittlement induced fracture is temperature. Although the overall mechanism of GB segregation is found to be almost consistent as a function of temperature, the fracture toughness and brittleness are temperature dependent (Ramasubramanian and Stein 1973). Relative orientation of GB with respect to advancing crack can affect crack tip stress distribution and ultimately determine the nature of transgranular vs. intergranular crack propagation. The analyses in this study, therefore, focus on characterizing effect of GB strength and GBs’ geometrical orientation with respect to crack propagation path on the resulting polycrystalline W fracture.

Fig. 1
figure 1

a A schematic showing HRTEM image of polycrystalline Ni doped W GB (Gupta et al. 2007), b atomic structure of W GB analyzed by Lee and Tomar Lee and Tomar (2013, 2014). Dimensional settings for the finite thickness GB interface samples and polycrystalline W samples are also shown

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2 Formulation

The overall focus of the present work is on understanding the influence of GB strength on crack propagation resistance. A 3-D GB interface model of Ni-doped W GB is developed for crack propagation simulations using XFEM Fig. 1. The GB simulation model is constructed based on the image of a realistic Ni doped W GB from a HRTEM image (Gupta et al. 2007) Fig. 1. In the case of polycrystalline W, this is the only explicit image available in scientific literature. Fully saturated Ni-doped W GBs have an average thickness of about 0.6 nm (Gupta et al. 2007). However, when calculating the GB properties one must insert a few atoms on either side of GB Fig. 1, resulting in an interface structure. During calculation of Ni doped W GB mechanical properties, Lee and Tomar (2013, 2014) followed such an approach and found that the minimum thickness of such an interface should be at least 2 nm in order to predict thickness independent maximum GB tensile strength. Thus, the minimum realistic thickness of GB interface in a FEM model of polycrystalline W material is 2 nm. In order to sufficiently resolve a GB interface model such as the one shown in Fig. 1 using finite elements, the sample size has to be in the similar size scale. Therefore, the sample size for the interface FE model is kept at 60 nm by 50 nm with finite GB thickness in the mid-sample location and 20 nm initial crack length, Fig. 1. Load is applied from upper and lower surface of the sample as shown in Fig. 1. Even though a realistic interface model has GB thickness of 2 nm, doubts persist whether at such a length scale continuum elements could be employed to enable FE model based predictions. Various formulations for predicting deformation of nanoscale structures using finite element method have been reported in literature, e.g. Guo et al. (2008). Earlier, a quantum mechanical ab-initio calculation framework based on the Car-Parrinello molecular dynamics (CPMD) simulation method has been used for calculating tensile strength of Ni doped W GBs by the authors (Lee and Tomar 2013, 2014). Stress–strain relation were obtained as a function of Ni percentage within GB region that could be used to calculate elastic-plastic material behavior parameters of the GBs. Since the relative size of the analyzed sample is in the same range as the GB thickness and since the properties of GB as well as grains on both sides of GBs are derived based on molecular simulations, the constitutive models incorporated in this work are assumed to incorporate the complexity associated with length scale effects.

The strain rate in CPMD based curves cannot be calculated as the stress for each strain is calculated for energy minimized structures (Lee and Tomar 2014). Current set of available state-of-the-art experiments cannot measure in-plane stress–strain curves GBs as a function of chemistry changes. Even classical molecular simulations fail at that. The CPMD simulation data used in the current work is the only choice. When one uses strain gradient plasticity to represent small scale physics one is trying to predict material behavior at nanoscale. The strain gradient model is trying to predict the small scale physics by parameter fitting. Situation here is different. We already have correct physics based stress–strain behavior from molecular simulations based on CPMD. The generalized plasticity model is only used to reproduce the CPMD simulation predicted behavior. The grain surface orientation on either side of GBs is kept at (100) and (001) in order to incorporate the influence of orientation change. Various other orientations were also tried. However, the effect of GB strength was the primary denominator that affected crack propagation resistance in the examined samples. In order to understand the influence of GB thickness, samples with three different GB thicknesses: 2, 4, and 8 nm are analyzed, Fig. 1. In order to understand and analytically predict the crack propagation in a polycrystalline microstructure, influence of GB orientation with respect to advancing crack on the resulting manner of intergranular or transgranular crack is analyzed Fig. 1. GB strength is varied as a function of GB chemistry based on earlier work of Lee and Tomar (2013, 2014). The sample thickness values were arrived at by performing fracture toughness calculations as a function of specimen thickness. The value chosen was finally the value that lead to thickness independence. The simulations are performed in the plane strain condition.

Polycrystalline W microstructure samples size analyzed is also shown in Fig. 1. The orientation of grains in the examined W microstructures is randomly varied among [100], [010], and [001] as shown later. During crack propagation simulations through W microstructures, finite GB width cannot be considered due to a large difference between size scales of grains and GBs. Therefore, while simulating crack propagation through polycrystalline W microstructures a framework based on a combination of XFEM with the cohesive finite element method (CFEM) is used.

Table 1 Stress and strain parameters of grains and an example grain boundary (25 at.% Ni)
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CFEM is exclusively used to represent interface separation properties of W GBs with peak strength predicted in earlier works by authors (Lee and Tomar 2013, 2014). The finite element mesh in the case of both interface and polycrystalline geometries has been generated in an unstructured manner. The convergence study was performed in all the cases in terms of crack path and fracture energy in order to establish the appropriateness of the FE mesh used. No re-meshing technique has been used once the crack propagation simulations start since XFEM discretization allows the model to add degrees of freedom for those cracked elements with enrichment. One important issue is orientation dependence of grain properties in the polycrystalline W FEM model. In this work, all grains are modeled with property calculated based on [100], [010], and [001] oriented W crystal deformation Table 1.

These property values are fitted to a generalized plasticity model described in next section. The fitting is implicitly performed usingin ABAQUS that already has material model implemented. Since ABAQUS software is used (as mentioned on page 11) one does only need to prescribe parameters listed in Table 1 for the isotropic hardening law already implemented in ABAQUS. The Table also shows parameters for a GB with 25 % atomic wt% Ni. These GB property values are lifted directly from author’s earlier works (Lee and Tomar 2013, 2014). It is important to mention that while the calculations of Lee and Tomar (2013, 2014) focused on a range of Ni atomic fractions (e.g. saturation levels) the Table shows only 25 atomic Ni fraction. Later, in the work parameters corresponding to the other atomic fraction values are used, which are not listed here for brevity.

2.1 Constitutive model

For the constitutive description of grains and GBs, a general elastic-plastic material model is used to explicitly represent the mechanical response. In this research, standard rate independent linear isotropic hardening model is fitted to the molecular stress–strain data. Based on the rate-independent loading condition, a flow rule combined with the deviatoric elasticity has been used to integrate strain decomposition (Weber et al. 1990). The yield function is described as a function \(f_{\mathrm{i}} \left( {\upsigma ,H_{\mathrm{i},\upalpha } }\right) \), where \(H_{\mathrm{i},\upalpha } \) is the hardening parameter described in reference (Cermelli et al. 2001) and \(\upsigma \) is effective stress. The material model is already implemented as a standard model in ABAQUS/standard (2010). The elastic part of the response is assumed to be derivable from an elastic strain energy density potential (ABAQUS Users Manual 2010), so the stress is defined by,

$$\begin{aligned} \sigma =\frac{\partial U}{\partial \varepsilon ^{el}}. \end{aligned}$$
(1)

Here, U is the strain energy density potential. Basic assumption of the model is that the deformation can be divided into an elastic part and plastic part (Agah-Tehrani et al. 1987). In the most general form the deformation gradient based on multiplicative decomposition (Lee 1969), is

$$\begin{aligned} F=F^{el}F^{pl}. \end{aligned}$$
(2)

Here, \(F^{el}\) is the fully recoverable part of the deformation and \(F^{pl}\) is the part that cannot be recovered. The rigid body rotation at a point can be included in the definition of either \(F^{el}\) or \(F^{pl}\) or can be considered separately before or after either part of the decomposition. From the definition of Lagrangian strain, the elastic and plastic finite strains are,

$$\begin{aligned} E^{el}=\frac{1}{2}\left( {F^{{el}^{T}}F^{el}-I} \right) \end{aligned}$$
(3)

and

$$\begin{aligned} E^{pl}=\frac{1}{2}\left( {F^{{pl}^{T}}F^{pl}-I} \right) , \end{aligned}$$
(4)

respectively. Since, the elastic strain in metals is usually small, an infinitesimal strain increment can be expressed as summation of elastic and plastic strain increment components. Therefore, in this work infinitesimal strain formulation is used.

2.2 XFEM formulation

The basic concept of XFEM is based on describing nodal and element displacement fields as a combination of a continuous part \(\mathbf{u}^{\textit{FE}}\) and a discontinuous part \(\mathbf{u}^{\textit{enr}}\). In addition, a jump function is added to allow the presence of discontinuities in an element by enriching degrees of freedom. The formulation of displacement field in XFEM is expressed as

$$\begin{aligned} \mathbf{u}^{\mathrm{h}}(\hbox {x})= & {} \mathbf{u}^{\mathrm{FE}}+\mathbf{u}^{\mathrm{enr}}\nonumber \\= & {} \sum _{\mathrm{j}=1}^{\mathrm{n}} {\hbox {N}_{\mathrm{j}} (\hbox {x}) \hbox {u}_{\mathrm{j}} } +\sum _{\mathrm{k}=1}^{\mathrm{m}} {\hbox {N}_{\mathrm{k}} (\hbox {x})\hbox {H}(\hbox {x})} \hbox {a}_{\mathrm{k}} . \end{aligned}$$
(5)

Here, N is shape function, u is nodal displacement vector, H(x) is jump function or Heaviside function, and a is nodal enriched degree of freedom vector. For the crack initiation criteria, the maximum principal stress criterion has been used. This indicates that the crack will propagate from its tip in a direction of \(\uptheta \) of maximum principal stress.

2.3 XFEM with CFEM

XFEM is a type of strong discontinuity method which takes account of discontinuities within an element using partition of unity (Belytschko and Black 1999; Moës et al. 1999; Dolbow et al. 2000). However, interfacial crack that segregates GBs cannot be simulated using XFEM when GBs modeled are infinitesimally thin in comparison to the grains. Such is the case when crack propagation through polycrystalline W is modeled in this work. Therefore, Failure in GBs is represented by segregation of element surfaces based on surface-based cohesive behavior. The surface cohesive traction is based on maximum tensile strength (Tomar et al. 2004), calculated for GBs as a function of chemistry by Lee and Tomar (2013, 2014). The cohesive law is bilinear with the evolution of damage defined using the Benzeggagh–Kenane (BK) form with exponent \(\upeta \) of 1.75 (Benzeggagh and Kenane 1996). The BK damage evolution criterion is

$$\begin{aligned} \textit{G}_\textit{C} =\textit{G}_{\textit{IC}} +(\textit{G}_{\textit{IIC}} -\textit{G}_{\textit{IC}} )\left( {\frac{\textit{G}_{\textit{Shear}} }{\textit{G}_T }} \right) ^{\eta }, \end{aligned}$$
(6)

where \(\textit{G}_{\textit{shear}}=\textit{G}_{\textit{II}}+\textit{G}_{\textit{III}}\), \(\textit{G}_{T}=\textit{G}_{I}+\textit{G}_{\textit{II}}+\textit{G}_{\textit{III}}\). Here “I”, “II”, and “III” subscripts refer to opening, shear and tear fracture modes, respectively. G is used to denote fracture energy. Subscript “c” is for critical fracture energy. The Benzeggagh–Kenane damage evolution criterion is primarily used when the critical fracture energies during deformation purely along the first and the second shear directions are the same, i.e. \(\textit{G}_{\textit{IIC}}=\textit{G}_{\textit{IIIC}}\). In addition, since the present work simulations are performed in 2-D manner, such assumption is justified. In ABAQUS one needs to specify exponent \(\upeta , \textit{G}_{\textit{IC}}\), and \(\textit{G}_{\textit{IIC}}\). The fracture energy values are based on the GB separation energies calculated in the work of Lee and Tomar (2013, 2014) for quasi-2D GB structures Fig. 1 and Table 2.

Table 2 \(\textit{G}_{\textit{IC}}\) and \(\textit{G}_{\textit{IIC}}\) values
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\(\textit{G}_{\textit{IC}}\), and \(\textit{G}_{\textit{IIC}}\) are found to be the same in this work. The exponent value of 1.75 is calculated based on reference mode-I and II crack propagation tests performed on monolithic samples in plane strain and the corresponding convergence in the SIF data. At a point in microstructural crack propagation simulation, all elements with crack tip are traced. If the crack tip is in a grain, XFEM is used to determine crack propagation condition. If the crack tip is at interface, CFEM is used. In addition, stresses at interfaces are monitored along with damage evolution in order to determine if the crack starts branching along GBs. Although there are two damage models that exist in a single microstructure when both XFEM and CFEM are used, the simulation approach follows crack tip and is designed to follow only one damage model at a single simulation increment. Thus, it could flexibly represent either inter-granular (CFEM) or trans-granular (XFEM) crack propagation.

2.4 Brittleness index calculation

As hardness H measures resistance to deformation and fracture toughness \(K_c \) measures resistance to fracture, it is reasonable to consider the comparative values of these two parameters as an indicator of brittleness. The commonly used BI parameter to identify extent of brittleness of a material (Lawn and Marshall 1979; Evans and Wilshaw 1976), is

$$\begin{aligned} \hbox {BI}=\frac{\hbox {H}}{\hbox {K}_\mathrm{c} }. \end{aligned}$$
(7)

The hardness of materials is usually obtained by indentation or scratch experiment. However, it is difficult to measure the hardness of W GBs due to small length scale. It has been found that the hardness of Ni doped W alloys is closely related to the maximum tensile strength. The maximum tensile strength for Ni doped W GBs is described as Lee and Tomar (2014),

$$\begin{aligned} T_{\max }= & {} T_{\textit{ideal}} \frac{\textit{CE}}{\textit{CD}}\frac{1}{\Phi } \left( {e^{n/10}-\frac{t}{t_{\textit{melt}} }+6.5} \right) \nonumber \\&\left( {1-10^{9}w+\left( {10^{9}w} \right) ^{2}} \right) . \end{aligned}$$
(8)

Here, \(T_{\textit{ideal}} \) is the pure W’s peak tensile strength, CE is atomic level cohesive energy, CD is the total GB cross-sectional area, \(\Phi \) is surface energy, n is Ni percentage, t is temperature, and w is GB thickness in m normalized by a critical length scale factor of \(10^{-9}\) m. A basis of development of this relation is shown in reference (Lee and Tomar 2014). Based on relation between hardness and fracture strength in the case of Ni doped W alloys (ASM International 1986), the following relation is used in this work:

Fig. 2
figure 2

Calculation of stress intensity factor at vicinity of crack when the crack-tip is located at a initial crack versus b at GB region, and the corresponding fracture toughness based on c stress intensity factor, and d J-integral when the crack is in GB region, as shown

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$$\begin{aligned} H=-1.823\times 10^{-10}T^{2}+2.51T+8.719\times 10^{8}. \end{aligned}$$
(9)

Owing to small scale of GBs hardness data for GBs is not available. This work uses the above relation for W alloys in the case of examined W GBs as well. Under the small scale yielding assumption, the SIF is calculated as Cartwright and Rooke (1975), Rooke et al. (1976)

$$\begin{aligned} K_I =\mathop {\lim }\limits _{r\rightarrow 0} \left[ {\sigma _{\textit{yy}} \left( {2\pi r} \right) ^{1/2}} \right] . \end{aligned}$$
(10)

Here, \(\sigma _{yy} \) is the stress component in loading direction near the vicinity of a crack, and r is the distance between crack tip and a point where stress is obtained (Steinmann et al. 2001). The small scale yielding is in line with plane strain conditions which were imposed based on fracture toughness measurements as a function of sample thickness before zeroing on to final sample thickness. Simple methods of calculating the SIF while having various types of discontinuity have been introduced (Rooke et al. 1981). Original formulation of the SIF has additional terms, including \(\uptheta \) in higher order adjunct to the equation above. However, the presented model has symmetrical loading and such conditions makes \(\uptheta =0\), which removes the rest of the terms. A series of methods for calculating the SIF have been reviewed in reference (Zhu and Joyce 2012). The present work uses the approach described in reference (Zhu and Joyce 2012) in order to explicitly derive SIF right from the FE results. J-integral calculation follows contour integral calculations along with an account of residual stresses as implemented in ABAQUS. Fracture toughness numerically is defined as the value of either SIF or J-integral corresponding to onset of crack propagation Fig. 2. Figure 2 shows fracture toughness calculated based on SIF and J-integral. As shown, quantitatively SIF based fracture toughness provides a conservative estimate of material fracture strength. There is no difference in the variation of fracture toughness as a function of Ni-percentage between those based on SIF and those based on J-integral.

The analyses reported in Fig. 2 were performed for larger sample sizes and at GB the reported fracture toughness values did not stiffer as a function of sample size. It is important to note that GB fracture toughness refers to fracture toughness of interface samples analyzed. The crack tip is located just next to GB. In the numerical calculations various element sizes next to the crack tip were tried and the convergence in mesh sizes dictated that the crack never crossed entire GB at once. The stress intensity factor is calculated when crack goes to the next element. Table 2 compares SIF based fracture toughness values for W GB with 25 at.% Ni with those for experimental polycrystalline samples (Table 3). Increasing GB thickness of the same sample size would lead to higher stress concentration at the crack tip near GB leading to increase in fracture toughness with increase in GB thickness (Table 3).

Table 3 Fracture toughness of W and W alloys in experimental study
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Fig. 3
figure 3

Effect of GB thickness on the a damage dissipation energy and b plastic dissipation energy of the 3-D model

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3 Results and discussions

As discussed earlier, the present work focuses on analyzing two factors that influence the crack propagation through W GBs. The first is the GB thickness which can be controlled by the level of saturation during the sintering process (Lee and Tomar 2013, 2014). In order to characterize the fracture of W GBs, parameters such as fracture toughness needs to be defined. The second factor is the geometric orientation of GBs with respect to advancing crack. In the following, crack propagation in polycrystalline W is analyzed based on an account of both factors. Afterwards, the results are correlated to the findings regarding crack propagation in polycrystalline W samples.

3.1 Effect of GB interface thickness and GB chemistry

As discussed earlier (Lee and Tomar 2013, 2014), found that the minimum thickness of an interphase that contains GB and few atomic layers around GB should be at least 2 nm in order to predict thickness independent properties of GB in ab initio simulations. In order to understand the influence of GB thickness on simulation results, crack propagation through interface samples with GB thickness of 2, 4, and 8 nm was analyzed. As shown in Fig. 3, change in thickness of GB does not significantly influence either the crack tip position change or crack tip energy dissipation.

Fig. 4
figure 4

Effect of Ni atomic wt% change on the a crack tip position change and b plastic dissipation energy with respect to the time-step increment

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Fig. 5
figure 5

Fracture toughness as a function of interface thickness and Ni at.% in the case of a initial crack tip position and b crack tip at the GB

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As shown by Lee and Tomar (2013, 2014) a little change in GB chemistry can significantly affect GB mechanical properties. The influence of such on crack propagation and fracture strength is analyzed by imparting the samples with 2 nm thick GB, the chemistry dependent mechanical properties based on data in the work of Margevicius et al. (1999), Ray et al. (1994). In their work, it has been found that the Ni atoms tend to segregate along the GB region and cause GB embrittlement as Ni impurity is added. As shown in Fig. 4, the crack speed as it passes through GB is a strong function of GB strength. The damage dissipation energy also follows the same pattern as crack tip position change. However, all the variations in crack tip position and damage dissipation energy are in a very narrow variation zone of 4–5 %.

3.2 Interfacial fracture toughness and brittleness index calculation

Obtaining the fracture toughness experimentally requires specific geometric and loading conditions including the specimen size which has to be large enough to ensure plane strain loading condition. However, requirements from fracture toughness test can’t be applied to the examined interface models in this work due to the limitation of length scale of analysis in current study. Based on the usual notion that the critical value of the SIF is equivalent to the fracture toughness, the fracture toughness of examined interface structures is obtained. Two values of fracture toughness are considered for each interface structure: (1) The fracture toughness at the initial crack tip position and (2) fracture toughness when the crack tip position is in the vicinity of GB. Both fracture toughness values corresponding to each Ni atomic % fraction are shown in Fig 5. As discussed earlier, the fracture toughness based on SIF provides a conservative estimate. As shown, the GB thickness and Ni atomic percentage change does not significantly influence interface fracture toughness.

3.3 Effect of length scale on brittleness index

Earlier, Mai and Lawn (1987) have used theoretical fracture mechanics based approach to calculate scale dependent fracture toughness (Mai and Lawn 1987).

Fig. 6
figure 6

Brittleness index with respect to the Ni at.% in GB for different interface thicknesses, a using J-Integral, and c using SIF. Flat black line respective average BI for grains

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Fig. 7
figure 7

A comparison of the revised BI for examined interface samples at all three simulation length scale as a function of Ni percentage using a J-integral and b using SIF

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Fig. 8
figure 8

Effect of GB angle with respect to advancing crack on the crack propagation path in 2 nm thick interface sample with orientation angle of a \(90^{\circ }\), b \(80^{\circ }\), c \(70^{\circ }\), d \(60^{\circ }\), e \(50^{\circ }\), f \(40^{\circ }\), g \(30^{\circ }\), h \(20^{\circ }\), and i \(10^{\circ }\) for GB with maximum tensile strength \(\hbox {T}=3700\) MPa

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Theoretical calculations showed that the fracture toughness has an intrinsic length-scale dependence. In the work of Mai and Lawn (1987), the range of fracture toughness was found to be 2–7 MPa-m\(^{1/2}\) for Al. BI, which is defined as a ratio between the hardness and the fracture toughness of a material, can be calculated based on hardness values obtained using relation in Eq. (8) and fracture toughness values reported in Fig. 6. The obtained BI of GBs is compared with average BI of grains in order to characterize the GB embrittlement caused by Ni addition in Fig. 6. As shown, BI for GB samples differs significantly from the grain BI. An increase in Ni atomic wt% leads to first an increase in brittleness and then decrease. In order to characterize the GB embrittlement of realistic polycrystalline W samples one must consider interfaces with nm thickness. As discussed earlier, the BI parameter has an inverse square root of length scale dependence that can be expressed as \(B=\alpha L^{-1/2}\), where L can be characterized as the characteristic size scale in the examined samples. Various choices exist for choosing L. Authors tried minimum sample dimension. However, the BI needs to be independent of sample size. The length scale needs to be extractable and applicable at multiple analyses size scales. Therefore, in this case, the size scale is taken as the characteristic crack length at GB. Parameter \(\alpha \) is a coefficient for describing weight of the length scale for a specific material. Using the proportionality factor \(B=\alpha L^{-1/2}\), a modified brittleness index is introduced as

$$\begin{aligned} \hbox {B}^{*}=\frac{\hbox {H}}{\hbox {K}_\mathrm{c} }\cdot \frac{1}{\alpha }\sqrt{L}. \end{aligned}$$
(11)

The modified BI is dimensionless. In the case of examined GBs \(\alpha \) is found to be 3.3 such that modified BI for interface samples approaches the value of average grain modified BI. Figure 7 shows revised BI for the examined interface samples. The revised values also closely correspond to experimental W microstructure values reported earlier by Evan, Marshall, and co-workers (Lawn and Marshall 1979; Evans and Wilshaw 1976). Impurities in GB were known to be a main reason of embrittlement effect of GBs, but quantitative support was not available. From this comparison, it is clear that the nickel percentages of 50–60 % leads to the highest GB embrittlement effect. This finding matches with the quantum mechanical calculation based finding of Lee and Tomar (2013, 2014).

Fig. 9
figure 9

Crack propagation path as a function of the orientation of GB with respect to the initial crack in the case of GB with maximum tensile strength of \(\hbox {T}=4\) GPa for a angular range of \(50^{\circ }\)–\(90^{\circ }\) for trans-granular failure and b angular range of \(10^{\circ }\)–\(40^{\circ }\) for inter-granular and mixed mode failure

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3.4 Role of GB orientation in crack propagation

Computational simulations have revealed earlier that orientation of a GB with respect to an advancing crack can significantly affect crack propagation though GBs. The orientation relates to resolution of crack tip stresses along GBs and determines intergranular (crack advancing along GB) or transgranular (crack passing through GB) nature of failure. Based on such consideration the analyses next describe crack propagation in 2 nm thick interface samples as a function of orientation of interfaces with respect to advancing crack. Figure 8 shows crack propagation through 2 nm thick interfaces as a function of the angle of interface with respect to advancing crack. Due to the geometric symmetry in problem with respect to the horizontal axis, only the GB angles of 0\(^{\circ }\)–90\(^{\circ }\) with respect to advancing crack are analyzed. As shown, nature of crack propagation changes from transgranular to intergranular with increase in the angle from 0\(^{\circ }\) to 90\(^{\circ }\). Higher GB orientation angle with respect to advancing crack shows more of transgranular failure while lower GB angle leads to intergranular failure. This result is also found to be strongly correlated to the maximum tensile strength of GBs as discussed later.

It is clear from Fig. 8 that there is a specific threshold GB angle that separates the intergranular and transgranular failure for a fixed GB maximum tensile strength. Exact crack path for the case of maximum GB strength of \(\hbox {T}=4\) GPa is shown as a function of GB angle change in Fig. 9. The figure shows the crack path as a function of x–y coordinate. The interface sample mid-point lies at \(\hbox {x}=30\) \(\upmu \)m. From the description shown in Fig. 9a, crack propagations through GB angles in the range of 50\(^{\circ }\)–\(90 ^{\circ }\) is in the form of transgranular failure type. In the case of transgranular failure, deflection of crack path appeared to be greater as GB angle decreases. The greater deflection indicates higher energy required for crack propagation. Intergranular failure and mixed mode failure occur for GB angles with respect to advancing crack varying in the range of \(10^{\circ }\)–\(40^{\circ }\).

3.5 Validation of the crack path change trend from experimental work

Prediction of crack propagation path has been studied experimentally by Watanabe and Tsurekawa (1999). It was found that the determination between intergranular and transgranular failure type for an advancing crack in a microstructure is strongly correlated with GB mechanical properties including the fracture strength. For polycrystalline molybdenum based alloy, properties of GBs were found to be controllable by impurity contents, processing method, and crystal structure. It was found that angular GB distribution (dispersion of GB angles with respect to X axis for all GBs in a microstructure) is strongly influenced by the material processing method, impurity contents, and GB embrittlement.

Considering that the fracture energy is generated from creation of fracture surfaces, fracture path has direct correlation with the dissipation of energy. From the analyses of crack propagation in 2 nm thick GB interface models, dissipation of internal energy was analyzed as a function of maximum tensile strength of GBs, Fig. 10. GB angles of \(80^{\circ }\) and \(10^{\circ }\) were chosen for plotting as those correspond to opposing trend of transgranular and intergranular failure, respectively, in all cases.

Fig. 10
figure 10

Total internal energy (elastic strain energy + plastic dissipation energy) as a function of the maximum tensile strength of interface samples with 2 nm thick GB in the case of a GB at an angle of \(80^{\circ }\) with respect to advancing crack, and b GB at an angle of \(10^{\circ }\) with respect to advancing crack

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From the internal energy dissipation results as a function of the GB angle, it is found that a lower GB angle with respect to advancing crack leads to higher variation in fracture energy dissipation. Almost invariably, the change in maximum GB tensile strength does not affect internal energy dissipation in the case of GBs oriented such that the crack path is transgranular.

3.6 Modified brittleness index as a function of microstructure

Figure 11 shows revised BI as a function of GB angle based on Eq. (11). Figure 11a shows revised BI for maximum GB tensile strength of 3600 MPa. Figure 11b shows revised BI as a function of GB maximum tensile strength for different inclinations of GBs with respect to advancing crack. The results indicate a strong dependence of revised BI on GB angle as well as on maximum GB tensile strength which is influenced by fundamental GB chemistry. Results indicate that the GB level embrittlement is influenced by two different aspects: geometric orientation of GB with respect to advancing crack and strength property of GBs.

Fig. 11
figure 11

Revised brittleness index a as a function of GB angle with respect to initial crack in the case of GB with maximum tensile strength of 3600 MPa and b as a function of maximum GB tensile strength for different inclinations of GBs with respect to advancing crack

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As shown in Fig. 11, the BI decreases at certain GB angles for a given GB strength. However, higher strength property of GBs usually leads to higher value of BI Fig. 11b. Some GB angles remain insensitive to maximum GB tensile strength increase. For example, a GB angle of \(50^{\circ }\)–\(70^{\circ }\), has relatively lower BI at all GB maximum tensile strength values, which implies that in such GB structure the possibility of having GB embrittlement during failure is low. While brittleness of bi-granular GB/interface crack propagation models analyzed here can be generalized as a function of geometric inclination of GB with respect to advancing crack, the polycrystalline material model makes it difficult to find a correlation between GB geometric orientation and BI due to higher number of GBs with varied orientations with respect to advancing crack. In order to analyze this issue, BI of three different microstructure images picked from literature was analyzed. As shown in Fig. 12, their BI values range in a narrow band. Next section, describes microstructural crack propagation studies in more detail.

Fig. 12
figure 12

Effect of microstructure morphology on the brittleness index with respect to three different polycrystalline W microstructures analyzed

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Fig. 13
figure 13

a Failure index as a function of GB angle with respect to initial crack and maximum GB interface tensile strength, and b failure index prediction criteria derived from the developed relation in Eq. (11)

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Fig. 14
figure 14

Two different grain orientation of the microstructure examined for applicability of FI criterion: a ORI-1, b ORI-2

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In order to accurately analyze and predict manner of intergranular vs. transgranular crack propagation in polycrystalline W microstructures, every GB angle along an advancing crack can be individually analyzed based on its maximum tensile strength and GB angle with respect to advancing crack. Such determination can be analytically performed based on numerical result obtained so far. Such strategy is pursued in the next section.

Fig. 15
figure 15

Crack propagation path as a function of the maximum GB tensile strength (T) and varying orientation of grains when a, b \(\hbox {T} = 3500 \hbox {MPa}\), c, d \(\hbox {T} = 3700 \hbox {MPa}\), e, f \(\hbox {T} = 3900 \hbox {MPa}\), and g, h \(\hbox {T} = 4100 \hbox {MPa}\)

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Fig. 16
figure 16

a Plastic energy dissipation and b total energy internal energy for microstructure fracture with different GB interface tensile strengths

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Fig. 17
figure 17

Comparison of crack path (black) in polycrystalline W with the predicted crack path (red) from failure index for different morphologies and different length scales (a, b, c, e, f)

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3.7 Development of a criteria to predict transgranular versus intergranular failure type

The present study includes failure simulations that consider the orientation of GB with respect to advancing crack in a range of \(0^{\circ }\) to \(90^{\circ }\)and the maximum tensile strength of GB in a range of 3500–4200 MPa. The range of angles can cover most possible GB orientations with respect to advancing crack in most polycrystalline materials. Transgranular and intergranular crack propagation represent two extremes of crack propagation modes in a microstructure. Most commonly, the crack propagation path in a polycrystalline microstructure is a combination of the two. If one had to represent transgranular and intergranular failure modes by numbers, values of 1 and \(-\)1 can be assigned with 0 being assigned for “perfect” mixed mode crack that spends 50 % along GB and 50 % across. Based on such number representation, a failure index (FI) can be developed using numerical data in this work that lies between 1 and -1. Maximum GB tensile strength and GB angle with respect to advancing crack can be fitted into such FI relation to predict crack propagation path through a polycrystalline microstructure.

FI is described as

$$\begin{aligned} \textit{FI}=a+b\frac{T_{\textit{GB}} }{T_{\textit{Grain}} }+c\theta ^{2}. \end{aligned}$$
(12)

Here, \(T_{\textit{GB}}\) is maximum tensile strength of GB, \(T_{\textit{Grain}}\) is maximum tensile strength for the grain, and \(\uptheta \) is the GB angle. In the case of polycrystalline W in this study, constants a, b, and c are 4.45, \(-\)4.2, and 0.00024, respectively. The FI values in Fig. 13a were fitted in Eq. (12) using a least square regreesional fit procedure to obtain the values of constants a, b, and c. A linear dependence on the strength ration and a quadratic dependence on the orientation angle was not accidental. Various forms of a combination of such strength and angle relationship were tried. In the end the relation in Eq. (12) predicted the correlation the best. One could physically interpret such a relation to indicate that crack path resistance has stronger influence on fracture than the material inherent strength. Figure 13b plots the FI as a function of \(T_{\textit{GB}}/T_{\textit{Grain}}\). By correlating the ratio of maximum tensile strength of GBs and grains with GB orientation with respect to advancing crack, the relation for FI considers both mechanical and geometrical property of GBs. A 2-D representation of FI is shown in Fig. 13b.

The crack propagation based failure in polycrystalline microstructure involves numerous grains and GBs. At a particular location, the crack tip tends to advance in the direction of the mechanically weakest point. In order to understand the applicability of relation in Eq. (11) to polycrystalline material failure, crack propagation simulation in sample polycrystalline W microstructures were performed as a function of change in maximum tensile strength of GBs and two different arrangements of grain orientations mentioned earlier in Table 1 and Fig. 14.

As shown in Fig. 15, change in maximum GB tensile strength (while all other factors being unchanged) significantly affects the change in crack propagation path in the examined microstructure. Initially perfect intergranular failure can be observed in Fig. 15a, b, e, f. With increase in GB tensile strength initial failure mode first turns into transgranular followed by a switch back to intergranular. No clear orientation dependence is observed. Besides main crack, slight crack branching is observed in call cases with random pattern. The FI criterion is used to track only the main crack.

Average plastic energy dissipation and internal energy dissipation as a function of maximum tensile strength for the examined microstructure are shown in Fig. 16a, b, respectively. Magnitude of both energy dissipations is in the similar range which implies that most of the energy is dissipated during plastic deformation. From Fig. 16, it can be observed that the GBs inclined at steeper angles (\({>}45^{\circ })\) produced high percentage of intergranular failure. The energy dissipation results shown in Fig. 16 show that lower energy dissipation occurs with lower maximum GB tensile strength. This result is quite intuitive considering the basic notion that crack propagation path tends to follow a route where it needs less energy to generate crack surfaces. However, such crack propagation path also leads to longer crack.

Figure 17 shows a comparison of crack propagation path prediction based on FI (red in color and grey in black and white) and numerically simulated crack path in 3 polycrystalline W microstructures (dark black line) with two different sets of orientation distribution of grains. Boundary conditions for crack propagation simulations are specified in Fig. 1. The crack path predictions are made based on Eq. (11) for given grain and GB strength values. As shown the crack path predictions based on Eq. (11) and calculations based on XFEM match with each other. In order to understand the influence of sample size, crack propagation paths were analyzed for larger sized structures as well. The results were found to be independent of sample size (not shown). This is primarily because the criterion being employed here locally at the crack-tip and orientation of local GB with respect to immediate crack tip.

4 Conclusion

Fracture of polycrystalline materials is known to have various failure types depending on microstructural morphology and individual constituent mechanical properties. GBs of polycrystalline materials have long been the topic of interest from the point of view of engineering microstructural properties. GB segregation is suspected to impart a significant hardening of the GB region relative to that of the grain interior. Such hardening of the GB region is directly correlated with the level of plasticity within a W microstructure which possibly causes GB embrittlement effect. A number of experimental studies have reported that both intergranular and transgranular fracture can be observed in polycrystalline W. One of the reasons for having inconsistent wide variation in varying fracture modes is known to be the impurity contents in W material.

In this study, crack propagation simulations in 3-D GB models for Ni doped W GBs are performed using XFEM. Stress–strain relations of Ni doped W GBs obtained from quantum mechanical simulations are used to build a representative elastic-plastic material model. Both geometric (orientation of GB w.r.t. to advancing crack) and chemical doping effects of GB are investigated in regard to their influence on crack propagation. While incorporating quantum mechanical GB properties in continuum simulation, an important issue is the thickness of GBs in the continuum polycrystal being simulated. The resulting analyses introduce a dimensional parameter in the BI in order to make BI predictions for GBs closer to experimental values. Using the modified BI, GB embrittlement is quantitatively analyzed as a function of GB chemistry in order to find GB chemistry corresponding to most severe GB embrittlement. Based on the findings of crack propagation path prediction in polycrystalline W, a quantitative criterion which predicts crack propagation path is proposed. The criterion is based on definition of FI which is shown to quantitatively approximate crack path in polycrystalline W microstructures analyzed in this work. Finally, in this work crystal plasticity type of simulations were not considered. Rather a generalized plasticity model was used. A whole new body of experiments is needed before performing crystal plasticity type of work for W similar to what is done in the case of Ni or Ti. This is one of the limitations of the performed work.