Modeling the Effect of Grain Boundary Segregations on the Fracture Toughness of Nanocrystalline and Ultrafine-Grained Alloys

Abstract

A theoretical two-dimensional (2D) model is proposed that describes the effect of grain boundary (GB) segregations on the fracture toughness of nanocrystalline or ultrafine-grained alloys. It is shown that GB segregations can lead to crack curvature, providing both crack surface roughness and crack deflection near the crack tip. Within the model, the growth of cracks along GBs under the action of a tensile load is considered. The effects of brittle GB segregations on the crack surface roughness and crack deflection near the crack tip are analyzed, and the associated increase in the fracture toughness of the material is calculated. It is shown that toughening can be achieved if segregations are very brittle and occupy a moderate proportion of GBs. In particular, a sufficiently large (up to 50%) fraction of GBs containing very brittle segregations can increase the fracture toughness by 30–35%. The results of the model can be applied to thin nanocrystalline or ultrafine-grained films.

1. Introduction

Due to their ultrahigh strength, nanocrystalline and ultrafine-grained metallic materials have been a subject of intensive research in the last few years (see, e.g., reviews [1,2,3,4,5,6]). At the same time, pure nanocrystalline metals are thermally unstable against grain growth. A way to increase the thermal stability of nanocrystalline and ultrafine-grained metallic materials is the fabrication of alloys containing grain boundary (GB) solute segregations [7,8,9,10,11]. In addition to providing thermal stability, GB solute segregations in nanocrystalline and ultrafine-grained alloys can affect the mechanical properties of such alloys. Experimental observations [12,13,14] demonstrated that such segregations can strongly increase the strength and hardness of these alloys. This is attributed to various factors, such as inactivation of GB dislocation sources [15,16,17], resistance to dislocation propagation across grains [18], and suppression of GB sliding [17,19].

Along with strength and hardness, GB segregations in nanocrystalline and ultrafine-grained alloys affect the fracture toughness of such alloys. Although GB segregations can, in some cases, reduce the fracture toughness of nanocrystalline alloys [20,21,22], recent experiments [22] with nanocrystalline Pt-Au alloys containing GB segregations of Au demonstrated the possibility of increasing the fracture toughness and ductility of such alloys due to the formation of segregations. The toughening effect of GB segregations in these alloys was observed at moderate (5 at.%) Au concentration, whereas at high (10 at.%) Au concentration, GB segregations of Au decreased fracture toughness. The toughening associated with the GB segregations of Au was attributed [22] to the formation of satellite cracks at GBs containing brittle GB segregations.

Since GB segregations are often characterized by weak atomic links and can serve as sites of easy fracture initiation and propagation, GBs containing such segregations can be considered weak (brittle) boundaries. Recent finite element (FE) simulations [23] of crack propagation along grain or interphase boundaries that form a two-dimensional hexagonal lattice within an elastic–viscoplastic constitutive model demonstrated the possibility of fracture toughness enhancement at a moderate (up to 10–40%) fraction of brittle boundaries followed by a drop in fracture toughness at a higher fraction of such boundaries. The authors of [23] attributed the toughening associated with brittle boundaries to the combination of various toughening mechanisms, such as crack deflection, crack branching, the formation of satellite cracks, and enhanced plastic deformation near the crack tip. However, which of these toughening mechanisms are most important for toughening remains unclear.

Similar effects of toughening associated with structural inhomogeneities have previously been studied in other FE simulations [24,25,26,27,28,29]. These simulations demonstrated in particular that the formation of elongated grains with GBs providing high angles with the macroscopic crack growth direction [25], the appearance of bimodal grain size distribution [24], or the presence of inhomogeneous residual stresses [27] increases fracture toughness.

The simulations of crack deflection associated with the geometry of GBs [25] or the presence of planar inclusions [30] point to the important role of this toughening mechanism in overall fracture toughness enhancement. One can assume, therefore, that toughening due to GB segregations is, to a large extent, associated with crack deflection due to the presence of brittle GBs (containing segregations) that make high angles with the macroscopic crack growth direction. The aim of this work is to conduct a theoretical study of the effect of crack deflection in nanocrystalline and ultrafine-grained alloys with GB segregations on the fracture toughness of such materials. Here, we consider a 2D (two-dimensional) model that is best suited for thin films with a thickness of one to several grain diameters. We focus on two toughening mechanisms associated with crack deflection: crack deflection in the wake of the crack, which leads to crack surface roughness, and crack deflection near the crack tip. These two toughening mechanisms will be considered in detail in the next sections.

2. Toughening Associated with Crack Surface Roughness

Consider a nanocrystalline metallic alloy in which a mode I crack grows under the action of a model tensile load ${\sigma }_0$ (Figure 1a). It is assumed that the crack mainly grows along grain boundaries (GBs), and the crack length is much greater than the GB length (Figure 1a). Consider the situation in which segregations of impurity atoms are randomly formed in some GBs, are located along the entire length of such GBs, and lead to a significant decrease in their effective surface energy (Figure 1a). The presence of brittle GBs with segregations can either lead to embrittlement of nanocrystalline alloys (if the fraction of GBs with segregations is large enough to form large clusters of such GBs, leading to catastrophic failure) or increase their fracture toughness due to the formation of satellite nanocracks in GBs with segregations and also as a result of crack deflection and/or branching. In this section, we consider the effects of segregation-related crack curvature on the fracture toughness of nanocrystalline and ultrafine-grained alloys. Such a curvature is determined by the possibility of crack propagation along the GB with segregations forming large angles with the macroscopic direction of crack growth, along which, in the absence of segregations, the crack would not grow.

Let us estimate the effect of crack deflection on the fracture toughness of the nanocrystalline or ultrafine-grained alloy. To do so, we introduce a rectangular coordinate system (x,y,z), as shown in Figure 1b, and represent stress ${\sigma }_{ij}$ acting on the solid with a crack (where i,j = x,y,z) as the sum of stress ${\sigma }_0 {\delta }_{iy} {\delta }_{iy}$, created by an external load in the solid without a crack and the additional stress ${\tilde{\sigma }}_{ij}$, which is necessary to satisfy the boundary conditions on the surface of the crack: ${\sigma }_{ij}={\sigma }_0 {\delta }_{iy} {\delta }_{iy}+{\tilde{\sigma }}_{ij}$. The angle between the tangent plane to the crack and the plane of the macroscopic crack direction is denoted as $\alpha$ (Figure 1b). Here, we focus on the model case, where $-\pi /2\le \alpha \le \pi /2$ everywhere at the crack surface. In this case, any value of the coordinate x corresponds to a single point at the crack surface characterized by angle $\alpha (x)$.

Introducing the normal and tangent directions n and τ, respectively, to the crack surface (Figure 1b), the boundary conditions on the crack surface S can be written as ${\left.{\tilde{\sigma }}_{nn}\right|}_S=-{\sigma }_0{\cos }^2\alpha$, ${\left.{\tilde{\sigma }}_{\tau n}\right|}_S=-{\sigma }_0\sin \alpha \cos \alpha$. Then the stresses ${\tilde{\sigma }}_{yy}$ and ${\tilde{\sigma }}_{xy}$ on the crack surface can be represented in the form

$${\left.{\tilde{\sigma }}_{yy}\right|}_S={\left.{\tilde{\sigma }}_{nn}\right|}_S{\cos }^2\alpha +2{\left.{\tilde{\sigma }}_{\tau n}\right|}_S\sin \alpha \cos \alpha =-{\sigma }_0{\cos }^2\alpha (1+{\sin }^2\alpha )\hspace{0.17em},$$

(1)

$${\left.{\tilde{\sigma }}_{xy}\right|}_S=-{\left.{\tilde{\sigma }}_{nn}\right|}_S\sin \alpha \cos \alpha +{\left.{\tilde{\sigma }}_{\tau n}\right|}_S\cos (2\alpha )={\sigma }_0{\sin }^3\alpha \cos \alpha \hspace{0.17em}.$$

(2)

Stresses ${\tilde{\sigma }}_{yy}(x,y)$ and ${\tilde{\sigma }}_{xy}(x,y)$ at the point $(x,y)$ of the crack surface $y=y_0(x)$ close to the flat surface $y=0$ can be expanded in a Taylor series in terms of the y-coordinate and thus expressed in terms of the stresses acting on the flat surface $y=0$:

$${\tilde{\sigma }}_{yy}(x,y=y_0(x))={\tilde{\sigma }}_{yy}(x,y=0)+y_0(x){\left.\frac{\partial {\tilde{\sigma }}_{yy}}{\partial y}\right|}_{y=0}+O\left(y_0(x){\left.\frac{\partial {\tilde{\sigma }}_{yy}}{\partial y}\right|}_{y=0}\right) \hspace{0.17em},$$

(3)

$${\tilde{\sigma }}_{xy}(x,y=y_0(x))={\tilde{\sigma }}_{xy}(x,y=0)+y_0(x){\left.\frac{\partial {\tilde{\sigma }}_{xy}}{\partial y}\right|}_{y=0}+O\left(y_0(x){\left.\frac{\partial {\tilde{\sigma }}_{xy}}{\partial y}\right|}_{y=0}\right) \hspace{0.17em}.$$

(4)

As a first approximation, only the first terms in the series can be taken into account, which yields:

$$\begin{gathered} {\tilde{\sigma }}_{yy}(x,y=y_0(x))\approx {\tilde{\sigma }}_{yy}(x,y=0)\hspace{0.33em}, \\ {\tilde{\sigma }}_{xy}(x,y=y_0(x))\approx {\tilde{\sigma }}_{xy}(x,y=0)\hspace{0.33em}. \end{gathered}$$

(5)

Thus, the problem of determining the stress intensity factor at the tip of a curved crack is reduced to the problem of finding the stress intensity factor at the tip of a flat crack characterized by stresses ${\tilde{\sigma }}_{yy}$ and ${\tilde{\sigma }}_{xy}$, which are defined by Equations (1) and (2), respectively, on the crack surface. Assuming that the angle $\alpha$ can be in the interval $-\pi /2\le \alpha \le \pi /2$, as a first approximation we model the crack characterized by stresses ${\tilde{\sigma }}_{yy}$ and ${\tilde{\sigma }}_{xy}$ oscillating along the crack surface as a crack characterized by constant stresses $<{\tilde{\sigma }}_{yy}(\alpha )>$ and $<{\tilde{\sigma }}_{xy}(\alpha )>$ averaged over angle $\alpha$. Since the angle $\alpha$ can take positive and opposite negative values with equal probability density and ${\tilde{\sigma }}_{xy}(\alpha ,y=0)$ is an odd function, we get $<{\tilde{\sigma }}_{xy}(\alpha ,y=0)>=0$. Thus, within our model, a curved crack is equivalent to a plane crack under the action of a constant tensile load ${\sigma }_0<s(\alpha )\hspace{0.17em}>$, where $s(\alpha )\hspace{0.17em}={\cos }^2\alpha \hspace{0.17em}(1+{\sin }^2\alpha )$ (see Equation (1)).

To calculate the effect of segregations on the fracture toughness of the nanocrystalline alloy, we use the energy criterion for crack growth. According to this approach, crack growth is possible under the condition:

$$F\ge 2{\gamma }_e$$

(6)

where F is the energy release, ${\gamma }_e=\gamma -{\gamma }_b/2$ is the effective specific surface energy, $\gamma$ is the specific surface energy, and ${\gamma }_b$ is the specific GB energy. For simplicity, it is assumed that all GBs with segregations are characterized by the same effective specific surface energy ${\gamma }_{e1}$ and all GBs without segregations are characterized by the same effective specific surface energy ${\gamma }_{e2}$.

A crack growing along GBs and stopped at a triple junction of GBs is examined (Figure 1a). Further growth of such a crack is possible along one of the two GBs adjacent to the triple junction. It is assumed that the crack will grow along the GB, for which the energy change $F-2{\gamma }_e$ associated with the growth of the crack per unit length is larger. To calculate the direction of crack growth, as a first approximation, we model a real curved crack as a flat mode I crack. In this case, the stress intensity factors $k_I(\alpha )$ and $k_{II}(\alpha )$ in the direction forming an angle $\alpha$ with the crack plane are given by (e.g., [31]):

$$k_I(\alpha )=K_I{\cos }^3(\alpha /2)\hspace{0.17em}$$

(7)

$$k_{II}(\alpha )=K_I{\cos }^2(\alpha /2)\hspace{0.17em}\sin (\alpha /2)\hspace{0.17em}$$

(8)

where $K_I$ is the stress intensity factor in the direction coinciding with the crack plane. In the approximation of an elastically isotropic solid, the energy release rate in the direction forming an angle $\alpha$ with the crack plane is calculated as [32]:

$$F(\alpha )=\frac{1-\nu }{2G}\left(k_I^2(\alpha )+k_{II}^2(\alpha )\right)$$

(9)

where $\nu$ is Poisson’s ratio and G is the shear modulus. Substituting Equations (7) and (8) into Equation (9) yields: $F(\alpha )=[(1-\nu )/(2G)]\hspace{0.17em}{K}_I^{2}g(\alpha )$, where $g(\alpha )={\cos }^4(\alpha /2)$.

Let the crack stop at the triple GB junction in front of the GBs, which form angles ${\alpha }_1$ and ${\alpha }_2$ with the GB plane and is characterized by the effective specific surface energies ${\gamma }_{e1}$ and ${\gamma }_{e2}$. Then, the condition of crack growth along the GB, characterized by parameters ${\alpha }_1$ and ${\gamma }_{e1}$, takes the form $F({\alpha }_1)-{\gamma }_{e1}>F({\alpha }_2)-{\gamma }_{e2}$. Taking into account the expression for $F(\alpha )$, this yields $g({\alpha }_2)<g({\alpha }_1)+A$, where $A=4G({\gamma }_{e2}-{\gamma }_{e1})/[(1-\nu )K_I^2]$. For definiteness, it is assumed that ${\gamma }_{e2}\ge {\gamma }_{e1}$ and that angles ${\alpha }_1$ and ${\alpha }_2$ can vary within the intervals $0\le {\alpha }_1\le \pi /2$ and $0\le {\alpha }_2\le \pi /2$, respectively. First, the situation where $A\le 1-g(\pi /2)$ is considered. In this situation, the solution of the inequality $g({\alpha }_2)<g({\alpha }_1)+A$ can be represented in the following form: $f({\alpha }_1,A)<{\alpha }_2\le \pi /2$ if ${\alpha }_{1c}<{\alpha }_1<\pi /2$ and $0\le {\alpha }_2\le \pi /2$ if $0\le {\alpha }_1\le {\alpha }_{1c}$, where ${\alpha }_1={\alpha }_{1c}=2\arccos [{(1-A)}^{1/4}]$ is the solution to the equation $g({\alpha }_1)+A=1$, and ${\alpha }_2=f({\alpha }_1,A)=2\arccos [{({\cos }^4({\alpha }_1/2)+A)}^{1/4}]$ is the solution to the equation $g({\alpha }_2)=g({\alpha }_1)+A$. In the case where $A>1-g(\pi /2)$, the inequality $g({\alpha }_2)<g({\alpha }_1)+A$ holds for any ${\alpha }_1$ and ${\alpha }_2$ within the intervals $0\le {\alpha }_1\le \pi /2$ and $0\le {\alpha }_2\le \pi /2$.

Thus, the absolute value of angle $\alpha$ that a crack fragment in a given GB forms with the macroscopic direction of crack growth is equal ${\alpha }_1$ for $g({\alpha }_2)<g({\alpha }_1)+A$, and to ${\alpha }_2$ otherwise. As a consequence, the average value $<s(\alpha (A))\hspace{0.17em}>$ of the function $s(\alpha (A))$ can be represented in the following form:

$$<s(\alpha (A))\hspace{0.17em}>=\left\{\begin{array}{l}(2/\pi )\hspace{0.17em}H(\pi /2),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}A>1-\hspace{0.33em}g(\pi /2),\\ \frac{4}{{\pi }^2}\left({\displaystyle {\int }_0^{{\alpha }_{1c}}s({\alpha }_1)\hspace{0.17em}d{\alpha }_1\hspace{0.17em}{\displaystyle {\int }_0^{\pi /2}d{\alpha }_2+{\displaystyle {\int }_{{\alpha }_{1c}}^{\pi /2}s({\alpha }_1)\hspace{0.17em}d{\alpha }_1}\hspace{0.17em}{\displaystyle {\int }_{f({\alpha }_1,A)}^{\pi /2}d{\alpha }_2}}}\right.\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+\left.{\displaystyle {\int }_{{\alpha }_{1c}}^{\pi /2}d{\alpha }_1}\hspace{0.17em}{\displaystyle {\int }_0^{f({\alpha }_1,A)}s({\alpha }_2)\hspace{0.17em}d{\alpha }_2}\right),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}A\le 1-\hspace{0.33em}g(\pi /2)\hspace{0.33em},\end{array}\right.$$

(10)

where $H(\alpha )={\displaystyle {\int }_0^{\alpha }s({\alpha }_0)\hspace{0.17em}d{\alpha }_0=[20\hspace{0.17em}\alpha +8\sin (2\alpha )-\sin (4\alpha )]/32}$.

Taking internal integrals in Equation (10) yields:

$$<s(\alpha (A))\hspace{0.17em}>=\left\{\begin{array}{l}5/8,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}A>3/4,\\ 5/8+(4/{\pi }^2){\displaystyle {\int }_{{\alpha }_{1c}}^{\pi /2}[H(f({\alpha }_1,A))-s({\alpha }_1)\hspace{0.17em}f({\alpha }_1,A)]\hspace{0.17em}d{\alpha }_1},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}A\le 3/4\hspace{0.33em}.\end{array}\right.$$

(11)

In the case where ${\gamma }_{e2}={\gamma }_{e1}$, we have A = 0, and Equation (11) yields $<s(\alpha (A=0))>=5/8+2/{\pi }^2$.

Let us denote the fraction of GBs with segregations (equal to the ratio of the number of GBs with segregations to the total number of all GBs) as c. Then, the probability that two GBs in front of the crack tip in a triple junction have different effective surface energies (that is, one GB contains a segregation and the other does not) is given by $P=2c(1-c)$. For the case of GBs with different effective surface energies, the above relation $A=4G({\gamma }_{e2}-{\gamma }_{e1})/[(1-\nu )K_I^2]$ holds. The probability that these two GBs have the same effective surface energies equals $1-P$. In a situation in which the GBs have the same effective surface energies, we obtain $A=0$. Thus, we have:

$$<s(\alpha )\hspace{0.17em}>=Ph\left(A\right)+(1-P)\hspace{0.17em}h(0)\hspace{0.17em},$$

(12)

where $h(A)=<s(\alpha (A))>$.

The length of the projection of the curved crack onto the plane $y=0$ is denoted as $2l$. Then the stress intensity factor $K_I$ at the tip of the curved crack can be represented as [5] $K_I=\beta {\sigma }_0<s(\alpha )\hspace{0.17em}>\sqrt{\pi l}=K_I^0<s(\alpha )\hspace{0.17em}>$, where $\beta$ is the factor that takes into account local plastic deformation at the crack tip ($\beta \le 1$ ), and $K_I^0$ is the stress intensity factor at the tip of a flat crack of length $2l$, which follows as $K_I^0=\beta {\sigma }_0\sqrt{\pi l}$.

The growth of the curved crack is beneficial under the condition $K_I=K_{IC0}$, where $K_{IC0}$ is the critical value of the stress intensity factor determined by the formation of new free surfaces during crack growth. On the other hand, to a remote observer, the crack is under the action of an external load ${\sigma }_0$ and is characterized by the stress intensity factor $K_I^0$. Macroscopically, the crack growth condition takes the form $K_I^0=K_{IC}$, where $K_{IC}$ is the fracture toughness. Substituting the critical conditions for crack growth $K_I=K_{IC0}$ and $K_I^0=K_{IC}^r$ into the relation $K_I=K_I^0<s(\alpha )\hspace{0.17em}>$, we obtain $K_{IC}^r=K_{IC0}/<s(\alpha )\hspace{0.17em}>$.

Using the latter equality, the ratio of the fracture toughness $K_{IC}^r(A)$ of the nanocrystalline/ultrafine-grained alloy with segregations to the fracture toughness $K_{IC}^r(A=0)$ of a similar alloy without segregations is calculated. This relation is defined by the expression:

$${\delta }_1=\frac{<s(\alpha )\hspace{0.17em}>{|}_{A=0}}{<s(\alpha )\hspace{0.17em}>}\hspace{0.33em},$$

(13)

which, accounting for Equation (12), yields:

$${\delta }_1=\frac{h(0)}{Ph\left(A\right)+(1-P)\hspace{0.17em}h(0)}\hspace{0.33em}.$$

(14)

Although it is assumed that the crack grows predominantly along GBs, in the path of a crack growing along many GBs there are triple junctions with unfavorably oriented GBs, where crack growth along a GB is unfavorable and the crack continues to grow from the triple junction through the grain interior. Therefore, it is assumed that the catastrophic growth of a crack highlights the possibility of intragranular growth. The critical condition for crack growth through the grain interiors takes the form $F(\alpha =0)=2\gamma$, which, taking into account the relations $F(\alpha =0)=[(1-\nu )/(2G)]\hspace{0.17em}{K}_I^2$ (see above) and $A=4G({\gamma }_{e2}-{\gamma }_{e1})/[(1-\nu )K_I^2]$, yields:

$$A=\frac{{\gamma }_{e2}}{\gamma }\left(1-\frac{{\gamma }_{e1}}{{\gamma }_{e2}}\right)\hspace{0.33em}.$$

(15)

When the ratio ${\gamma }_{e1}/{\gamma }_{e2}$ of the effective surface energies for GBs with and without segregations changes from 0 to 1, the value of the parameter $A$ varies from 0 to ${\gamma }_{e2}/\gamma$. In what follows, we set ${\gamma }_{b2}=\gamma /2$, where ${\gamma }_{b2}$ is the specific energy of the GB without segregations. Since ${\gamma }_{e2}=\gamma -{\gamma }_{b2}/2$, the last relation gives: ${\gamma }_{e2}/\gamma =0.75$.

Using the above expressions, we calculate the dependences of the parameter ${\delta }_1$, which characterizes the ratio of the fracture toughness of a nanocrystalline or ultrafine-grained alloy with GB segregations to the fracture toughness of a similar alloy without segregations, on the fraction c of GBs with segregations and the ratio ${\gamma }_{e1}/{\gamma }_{e2}$ of effective surface energies. These dependences are shown in Figure 2. From Figure 2, it follows that the value of the parameter ${\delta }_1$ increases with an increase in the proportion of GBs with segregations from 0 to 0.5 and/or a decrease in the value of ${\gamma }_{e1}/{\gamma }_{e2}$. This implies that the maximum fracture toughness can be achieved when GBs with segregations are very brittle, that is, characterized by the minimum values of ${\gamma }_{e1}$. At the same time, the proportion of GBs with segregations should be sufficiently large but should not exceed the values at which such brittle boundaries form clusters, leading to catastrophic fracture. As follows from Figure 2, in this case, the presence of GBs with segregations can increase the fracture toughness by up to 13%.

Thus, in this section, we have proposed an analytical model that describes the effect of impurity segregation along GBs on the fracture toughness of nanocrystalline alloys. Within the model, the presence of brittle GBs with segregations forming large angles with the macroscopic direction of crack growth leads to crack curvature, which, in turn, increases the fracture toughness. The model predicts that, for maximum fracture toughness, GBs with segregations must be very brittle and their proportion must be large enough.

3. Toughening Due to Crack Deflection at the Crack Tip

In the previous section, we considered toughening associated with crack roughness promoted by GB segregations. At the same time, segregations can also facilitate crack deflection near the crack tip, helping the crack to advance along GBs that form high angles with the macroscopic direction of crack growth. If a crack propagates over such a GB, the mode I stress intensity factor of this crack decreases, thereby hindering crack growth over the next GB. To estimate the effect of GB segregations on the crack deflection near the crack tip, a model straight crack AB with a small kink BC that forms an angle $\alpha$ with the crack plane is considered (Figure 1c). The stress intensity factors $k_I(\alpha )$ and $k_{II}(\alpha )$ in the direction of the kink BC are given by Formulas (7) and (8), where $K_I$ is replaced by $K_I^0$. Now, it is assumed that the crack makes another kink CD that produces an angle $\beta$ with the direction BC, as shown in Figure 1c. At distances that are small compared to the distance BC from point C, the stress field created due to the curved crack ABC is equivalent to that created in a solid with a straight crack lying in the plane BC and characterized by the stress intensity factors $k_I(\alpha )$ and $k_{II}(\alpha )$. Then, the stress intensity factors $k_I^{(1)}(\beta ,\alpha )$ and $k_{II}^{(1)}(\beta ,\alpha )$ in the direction CD can be calculated as (e.g., [31])

$$k_I^{(1)}(\beta ,\alpha )=k_I(\alpha ){\cos }^3(\beta /2)\hspace{0.17em}+3k_{II}(\alpha ){\cos }^2(\beta /2)\hspace{0.17em}\sin (\beta /2)\hspace{0.17em},$$

(16)

$$k_{II}^{(1)}(\beta ,\alpha )=-k_I(\alpha ){\cos }^2(\beta /2)\hspace{0.17em}\sin (\beta /2)\hspace{0.17em}+(1/2)\hspace{0.17em}{k}_{II}(\alpha )\hspace{0.17em}(3\cos \beta -1)\hspace{0.17em}\cos (\beta /2).$$

(17)

If BC and CD characterize GBs, then Equations (16) and (17) can be used to analyze the onset of crack propagation over GB CD. As a first approximation, we assume that if the crack starts to propagate over GB CD, it will grow over the entire GB. Then, the energy release rate that characterizes crack propagation over GB CD can be written as:

$$F(\beta ,\alpha )=\frac{1-\nu }{2G}\left({\left(k_I^{(1)}(\beta ,\alpha )\right)}^2+{\left(k_{II}^{(1)}(\beta ,\alpha )\right)}^2\right).$$

(18)

From Equations (7), (8), and (16)–(18) and the critical condition of crack growth $F(\beta ,\alpha )=2{\gamma }_e$, one can calculate the critical stress intensity factor $K_I=K_{IC}^{\mathrm{defl}}(\beta ,\alpha )$ for crack propagation over GB CD. For definiteness, in the following, we consider a situation where $\alpha >0$. In addition, for simplicity, we consider a specified value of angle $\beta$, $\beta =\pi /3$, which seems to reasonably describe the characteristic angles of crack advance for $\alpha >0$. To calculate the fracture toughness associated with crack growth over strong and weak GBs, we consider the strongest GB along which crack propagation is most difficult. This case is realized if GB CD has no segregation and is characterized by the effective specific surface energy ${\gamma }_{e2}$. At the same time, if the solid contains segregations, the most difficult case for crack propagation also corresponds to the situation where GB BC contains a segregation. The reason is that in this case, GB BC can make a higher angle with the macroscopic direction of crack growth, which can hinder further crack advance of GB CD.

The highest value of angle α can be found with Equations (6)–(9) and the critical condition for crack growth through the grain interiors $F(\alpha =0)=2\gamma$ as $g(\alpha )>{\gamma }_e/\gamma$, where $g(\alpha )={\cos }^4(\alpha /2)$, as above. The latter relation can be rewritten as $\alpha <{\alpha }_c$, where ${\alpha }_c=2\arccos {({\gamma }_e/\gamma )}^{1/4}$. Let us estimate ${\alpha }_c$ for two different situations where GB BC is strong (contains no segregations) and weak (contains a segregation). If GB BC has no segregation, then we have ${\gamma }_e={\gamma }_{e2}$. Using the estimate ${\gamma }_{e2}/\gamma =0.75$ suggested above, we obtain ${\alpha }_c\approx 0.75$ ($\approx 43°$). If GB BC contains a segregation, then we obtain ${\alpha }_c=2\arccos {(0.75\hspace{0.17em}{\gamma }_{e1}/{\gamma }_{e2})}^{1/4}$. As in the previous section, it is assumed that angle α does not exceed $\pi /2$. Therefore, if ${\alpha }_c>\pi /2$, we put ${\alpha }_c=\pi /2$.

The case in which crack propagation is most difficult corresponds to the angle $\alpha ={\alpha }_m$ from the interval $0\le \alpha \le {\alpha }_c$, at which the energy release rate for crack propagation over GB CD is lowest, that is, at which the function $F(\beta =\pi /3,\alpha )$ has the lowest value within the interval $0\le \alpha \le {\alpha }_c$. This case characterizes fracture toughness $K_{IC}^{*}=K_{IC}^{\mathrm{defl}}(\beta =\pi /3,{\alpha }_m)$ associated with crack deflection near the crack tip, which does not account for the crack surface roughness examined in the previous section. The parameter ${\delta }_2=K_{IC}^{*}({\gamma }_e={\gamma }_{e1})/K_{IC}^{*}({\gamma }_e={\gamma }_{e2})$ defines the ratio of the fracture toughness of a solid with GB segregations to that of a similar solid without GB segregations. The dependence of ${\delta }_2$ on the ratio ${\gamma }_{e1}/{\gamma }_{e2}$ of the effective surface energies for GBs with and without segregations is presented in Figure 3. It is seen in Figure 3 that the fracture toughness associated with crack deflection near the crack tip starts to grow if GBs with a segregation are sufficiently weak (${\gamma }_{e1}/{\gamma }_{e2}<0.5$), which enables them to promote crack deflection with sufficiently high angles. At ${\gamma }_{e1}/{\gamma }_{e2}\approx 0.35$, the value of the maximum deflection angle ${\alpha }_c$ approaches $\pi /2$, and ${\alpha }_c$ does not increase with a further decrease in ${\gamma }_{e1}/{\gamma }_{e2}$. As a result, the normalized fracture toughness ${\delta }_2$ also does not increase with a decrease in ${\gamma }_{e1}/{\gamma }_{e2}$ at ${\gamma }_{e1}/{\gamma }_{e2}<0.35$.

As is seen in Figure 3, the value of ${\delta }_2$ does not depend on the fraction c of weak GBs. This is a result of using a 2D model in which fracture toughness is controlled by crack propagation over most unfavorable (for crack advance) GBs, so even a very small fraction of weak GBs can considerably toughen the solid. A similar result was obtained in 2D FE simulations of crack propagation in a hexagonal lattice of boundaries [23], where essential toughening was observed for the minimum simulated proportion of weak boundaries (10%). However, the result would be different if we considered a 3D (three-dimensional) ensemble of GBs, where the propagation of the crack front would require crack advance over both weak and strong GBs. In this 3D case, the effect of crack deflection on fracture toughness would depend on the fraction of weak boundaries and would probably be smaller than that in our 2D model. In addition, in real materials, even in the 2D case, the parameter ${\delta }_2$ should at least weakly depend on the fraction c of weak GBs due to an increase in the crack length during crack propagation over GBs (which is neglected in our model, which assumes a large initial crack length).

Now let us calculate the fracture toughness $K_{IC}$ assuming that both crack surface roughness and crack deflection near the crack tip can contribute to the toughening of the solid. In this case, similarly to the equality $K_{IC}^r=K_{IC0}/<s(\alpha )\hspace{0.17em}>$ obtained in the previous section, one can write $K_{IC}=K_{IC}^{*}/<s(\alpha )\hspace{0.17em}>$. As a result, from formula (13) and the relation ${\delta }_2=K_{IC}^{*}({\gamma }_e={\gamma }_{e1})/K_{IC}^{*}({\gamma }_e={\gamma }_{e2})$, the ratio $\delta =K_{IC}(A)/K_{IC}(A=0)$ of the fracture toughness of a solid with GB segregations to that of a similar solid without GB segregations $K_{IC}$ can be derived as $\delta ={\delta }_1{\delta }_2$.

The dependences of the parameter $\delta$ on the fraction c of GBs with segregations and the ratio ${\gamma }_{e1}/{\gamma }_{e2}$ of effective surface energies are illustrated in Figure 4. A sharp increase in the parameter $\delta$ at $c=0$ is the result of the model approximations, which lead to the independence of the parameter ${\delta }_2$ on c. Therefore, one can assume that, in reality, the curves should be smoother near point $c=0$. From Figure 4, it follows that the value of parameter $\delta$ increases with an increase in the proportion of GBs with segregations from 0 to 0.5 and/or a decrease in the value of ${\gamma }_{e1}/{\gamma }_{e2}$. This means that the maximum fracture toughness can be achieved when GBs with segregations are very brittle, that is, characterized by the minimum values of ${\gamma }_{e1}$. At the same time, the proportion of GBs with segregations should not exceed the values at which such brittle boundaries form clusters, leading to catastrophic fracture. As follows from Figure 4, the presence of GBs with segregations can increase the fracture toughness by up to 34%. For comparison, Figure 4 also illustrates the normalized fracture toughness obtained in Ref. [23] for the situations in which the tensile load is normal to the zigzag direction of the 2D hexagonal lattice of GBs. It is seen that our calculations predict somewhat higher values of the normalized fracture toughness than the FE results [23]. The reason is that our calculations account for the effect of the crack surface roughness, which is not taken into account in [23], whereas Ref. [23] accounts for the possible formation of secondary cracks in front of the crack tip, which can decrease fracture toughness, especially at a high-enough fraction of weak GBs.

4. Concluding Remarks

Thus, in the present paper, we have suggested a model that describes toughening due to GB solute segregations in nanocrystalline or ultrafine-grained films. The model is valid for situations in which embrittling GB segregations occupy part of GBs and their fraction does not exceed the values at which such brittle GBs form clusters, leading to catastrophic fracture. Within the model, GBs predominantly propagate along GBs, and weak (brittle) boundaries promote crack deflection, which leads to toughening. Although the model is two-dimensional, it can describe the behavior of thin nanocrystalline or ultrafine-grained films, e.g., columnar-grained structures. The model predicts that toughening can be achieved if segregations are very brittle and occupy a moderate proportion of GBs. In particular, a sufficiently large (up to 50%) fraction of GBs containing very brittle segregations can increase the fracture toughness by 30–35%. The results of the model correlate with the experimental observations [22] of Pt-Au alloys, where GB segregations of Au led to toughening.

It should also be noted that the results of the model can be applied to situations in which brittle boundaries form due to factors other than solute segregations. These may include brittle inclusions in GBs, such as graphene platelets, brittle amorphous films, or GBs embrittled due to the presence of hydrogen. For example, the results of the model confirm that crack deflection in ceramic/graphene composites (where ceramic/graphene interfaces can be considered weak boundaries) contribute to the experimentally documented (e.g., [33,34,35,36,37]) toughening of ceramics due to the addition of graphene. Thus, the presence of brittle phases at GBs does not necessarily lead to embrittlement but can instead toughen the material.