II. THE CONSTITUTIVE MODEL

Both the dislocations within coarse grains and nanograins contribute to the overall strain hardening through Taylor's theory, thus the flow stress can be expressed as:

(1)$${\sigma _{{\rm{flow}}}} = {\sigma _0} + M\alpha \mu b\sqrt {{\rho _{\rm{T}}}} + {\sigma _{\rm{b}}}\quad ,$$

here α, μ, b, and M are the empirical constant, the shear modulus, the burgers vector, and the Taylor factor, respectively. σ₀ is the lattice friction stress and σ_b represents the back stress for the kinematic hardening that comes from dislocation accumulation at the grain boundaries, and these dislocations are emitted from the nanocrack tips because of stress concentration. ρ_T refers to the total density of dislocations in coarse grains and the nanograins, determined by the Kocks–Mecking model.^{Reference Kocks and Mecking23} It can be obtained by using the following equation:

(2)$${\rho _{\rm{T}}} = {f_{{\rm{coarse}}}}{\rho _{{\rm{sc}}}} + {f_{{\rm{nano}}}}{\rho _{{\rm{sn}}}}\quad ,$$

where f _coarse and f _nano are the volume fraction of the coarse grains and the nanograins, respectively, ρ_sc and ρ_sn are the density of statistically stored dislocations in the coarse grains and nanograins, respectively.

To describe the constitutive behavior of the BNC materials, a dislocation density-related unified constitutive model is used to describe the density of statistically stored dislocations below. During the deformation process controlled by dislocation mechanism, the density of statistically stored dislocations ρ_s is determined by two dynamic processes: athermal storage of dislocations $\rho _{\rm{s}}^ +$ and annihilation of dislocations $\rho _{\rm{s}}^ -$. For a given overall strain, ρ_s will be given by

(3)$${\rho _{\rm{s}}} = \rho _{\rm{s}}^ + + \rho _{\rm{s}}^ - \quad .$$

According to the Kocks–Mecking model, the density of dislocations in grains obeys the evolution law of plastic strain. Capolungo et al.^{Reference Capolungo, Jochum, Cherkaoui and Qu24} proposed a model accounting for the athermal storage of dislocations and the dynamic recovery process controlled by the annihilation of stored dislocations. Based on their model, the relation between the density of dislocations (ρ) and ε can be expressed as follows:

(4)$${{\partial \rho } \over {\partial \varepsilon }} = M\left( {{k \over d} + {k_1}\sqrt \rho - {k_2}\rho } \right)\quad ,$$

where k = 1/b, k ₁ = ψ/b, and k ₂ = k ₂₀(ε^p/ε₀)^−1/n, ψ is the proportionality factor, d is the grain size, k ₂₀ and ε₀ are the constants, ε₀ is the reference strain rate, and n is the Taylor orientation factor, which is inversely proportional to the temperature.

The back stress induced by dislocation accumulation can be simply expressed as:

(5)$${\sigma _{\rm{b}}} = M\left( {{{\mu b} \over d}} \right)N\quad ,$$

where N is the number of dislocations that emit from the nanocrack tips. In the present work, we assume that the nanocracks evenly distribute in the NC matrix, as shown in Fig. 3. In our previous work,^{Reference Liu, Zhou, Wang, Zhang and Wang25} we proposed a model to study the effect of grain boundary sliding and dislocation emission from crack tip on fracture toughness of NC materials. When an external load is applied to a specimen with a crack, stress concentration occurs at the crack tip, and then the stress singularity appears. Such stress singularity is induced not only by the external load but also by the stress field of dislocations introduced around the crack tip. Dislocations change the local stress intensity factor at the crack tip from where it would be in the bare external stress field. Such concept on the interaction between crack and dislocations has been treated in terms of crack tip shielding by dislocations.^{Reference Song, Curtin, Bhandakkar and Gao26,Reference Liu, Zhou and Shen27} To simplify, we assume that all dislocations emitted from the crack tip would reach the opposite grain boundary. From the literature,^{Reference Liu, Zhou, Wang, Zhang and Wang25} we can know the relation between the critical stress intensity factor K _IC and θ, where θ is the angle between the direction of dislocation emission and crack propagation direction. As illustrated in Fig. 4, the critical intensity factor K _IC reaches the maximum value when the angle θ is approximately 70°. Therefore, we only consider the number of dislocations emitted from the crack tip of NC copper when the angle θ is 70°. N can be got from

(6)$$N = {\rho _{{\rm{crack}}}}{N^*}\quad ,$$

where N ^∗ is the maximum number of edge dislocations that can be emitted from the crack tip along one slip plane, ρ_crack is the density of the nanocracks, and it can be written as follows^{Reference Zhu, Shi, Lu and Lu20}:

(7)$${\rho _{{\rm{crack}}}} = {\rho _0}\left( {1 - \exp \left[ { - {{\left( {{\varepsilon \over {{\varepsilon _0}}}} \right)}^m}} \right]} \right)\quad .$$

FIG. 3.

Schematics of nanocracks in bimodal structure.

FIG. 4.

Dependence of critical crack intensity factor K _IC on the angle θ made by the boundary plane and crack growth direction.

ρ₀ is the reference density of nanocracks and m is a parameter that can be defined by^{Reference Xie and Gao28}

(8)$$m\log {\varepsilon \over {{\varepsilon _0}}} = D\log {{\pi {a_0}{\sigma ^2}} \over {{\alpha _0}K_{{\rm{IC}}}^2}}\quad ,$$

where α₀ is a dimensionless coefficient, a ₀ is the initial crack size, and K _IC is the critical stress intensity factor. D is the fractal factor and it can be written as:

(9)$$D = 3 + {{\log \left( {{f_{{\rm{coarse}}}}} \right)} \over {\log \left( {{{{d_{\rm{L}}}} \mathord{\left/ {\vphantom {{{d_{\rm{L}}}} {{d_{\rm{S}}}}}} \right. \kern-\nulldelimiterspace} {{d_{\rm{S}}}}}} \right)}}\quad ,$$

here d _L and d _S are the maximum and minimum grain sizes, respectively.

Fracture originated in crack, stillness, and balance or development of cracks are all dependent on the stress field near the crack tip. For a mode-I crack with the length of 2a, when it is under different stress states, fracture toughness should be obtained by the following expression:

(10)$${K_{\rm{I}}} = {\alpha _0}\sigma \sqrt {\pi a} \quad ,$$

here K _I denotes the stress intensity factor, α₀ is the geometry factor, which depends on geometry and loading forms of the crack object, it can be set to 1 for a infinite plate through cracks.

a is the crack size. According to the fracture criterion of material, when K _I arrives at the value of critical stress intensity factor K _IC, substituting K _IC = K _I into Eq. (10) will lead to the expression of the crack size

(11)$$a = {{K_{{\rm{IC}}}^2} \over {{{\left( {{\alpha _0}\sigma \sqrt \pi } \right)}^2}}}\quad .$$

According to the energy balance theory, in the process of crack growth, the elastic energy would be released. When the maximum strain energy release rate reaches a critical value, the crack will expand. Based on this theory, one can obtain the relationship between the critical stress intensity factor and the critical energy release rate by

(12)$${K_{{\rm{IC}}}} = \sqrt {{{E{G_{{\rm{cr}}}}} \mathord{\left/ {\vphantom {{E{G_{{\rm{cr}}}}} {{\alpha _0}}}} \right. \kern-\nulldelimiterspace} {{\alpha _0}}}} \quad ,$$

where G _cr is the critical energy release rate, and it can be expressed by the surface energy

(13)$${G_{{\rm{cr}}}} = 2{\gamma _{\rm{s}}}\quad ,$$

here γ_s is the surface energy.

Let us calculate the maximum number of edge dislocations N ^∗ that can be emitted from the crack tip along one slip plane. The equilibrium position of each dislocation is calculated from the force balance equation F _r = 0 on the r-axis, here k = 1, 2, …, N ^∗. The total force $F_r^k$ acting on the kth dislocation can be written in terms of the effective stress $\sigma _{r\theta }^{\rm{e}}\left( {{r_k},\;\theta } \right)$, where r _k is the distance from the kth dislocation to the crack tip, and it can be presented as: $F_r^k = b\sigma _{r\theta }^{\rm{e}}\left( {{r_k},\;\theta } \right)$. The effective stress $\sigma _{r\theta }^{\rm{e}}\left( {{r_k},\;\theta } \right)$ can be written as^{Reference Ovid'ko and Sheinerman19}

(14)$$\sigma _{r\theta }^{\rm{e}}\left( {r,\;\!\theta } \right) = \sigma _{r\theta }^{{K_{\rm{I}}}}\left( {r,\;\theta } \right) + \sigma _{r\theta }^{{\rm{im}}}\left( {r,\;\theta } \right) \!+ \mathop\sum\limits_{\scriptstyle j = 1 \hfill \atop \scriptstyle j \ne k \hfill} ^{N^*} {{\sigma _{r\theta }}\left( {{r_k},\;{r_j},\;\theta } \right)} \quad ,$$

where the stress $\sigma _{r\theta }^{{K_1}}\left( {r,\;\theta } \right)$ is created by the applied load near the crack tip on the kth dislocation; the image stress $\sigma _{r\theta }^{{\rm{im}}}\left( {r,\;\theta } \right)$ is associated with the presence of the crack free surface and corresponds to the image force induced by the crack; and the stress σ_rθ(r _k, r _j, θ) is created by the jth dislocation at the point (r _k, θ) and corresponds to the force exerted on the kth dislocation by the jth dislocation.

Following the Rice–Thompson criterion,^{Reference Rice and Thompson29} it assumes that the emission of the first dislocation occurs if this dislocation is repelled from the crack tip when the distance from the dislocation to the crack tip exceeds the dislocation core radius r ₀. Thus, the critical condition for the emission of the first dislocation is

(15)$${\left. {\sigma _{r\theta }^{{K_{\rm{I}}}}\left( {r,\;\theta } \right) + \sigma _{r\theta }^{{\rm{im}}}\left( {r,\;\theta } \right)} \right|_{r = {r_0}}} > 0\quad .$$

For definiteness, we will consider only necessary conditions for dislocation emission and assume that dislocation emission can be possible even if the width of the crack tip attraction zone exceeds the dislocation core radius. We assume that the (N ^∗ + 1)th dislocation (N ^∗ = 1, 2, …) can emit within the interval 0 < r < d where this dislocation is repelled from the crack tip. In this region, the following inequality should be valid:

(16)$$\sigma _{r\theta }^{{K_{\rm{I}}}}\left( {r,\;\theta } \right) + \sigma _{r\theta }^{{\rm{im}}}\left( {r,\;\theta } \right) + \mathop\sum\limits_{\scriptstyle j = 1 \hfill \atop \scriptstyle j \ne k \hfill} ^{{N^*}} {\sigma _{r\theta }^{}\left( {{r_k},\;{r_j},\;\theta } \right)} > 0\quad .$$

It should be noted that Eq. (16) gives only necessary but not sufficient condition for the dislocation emission. Therefore, the real conditions for the dislocation emission are more complex than those described in Eq. (16). The stresses $\sigma _{r\theta }^{{K_{\rm{I}}}}\left( {r,\theta } \right)$, $\sigma _{r\theta }^{{\rm{im}}}\left( {r,\;\theta } \right)$, and σ_rθ(r _k, r _j, θ) appearing in Eqs. 14–16 can be obtained from Ref. Reference Lin and Thomson30.

To calculate the maximal number N* of lattice dislocations that can be emitted along the same slip plane, we use the following calculation procedure as introduced by Ovid'ko and Sheinerman.^{Reference Ovid'ko and Sheinerman31} First, we verify the validity of criterion (15) for the emission of the first dislocation. If this criterion is valid, we place the first dislocation at the distance d from the crack tip and verify the validity of criterion (16) for the emission of the second dislocation. If this criterion is valid for the second dislocation, we calculate its equilibrium position and check the validity of criterion (16) for the emission of the third dislocation, and so on. The procedure is carried out for all the new emitted dislocations and ends when criterion (16) for the emission of a new dislocation stops to be valid. Then, we can obtain the relationship between N ^∗ and the grain size d, as shown in Fig. 5.

FIG. 5.

The maximum number N* of dislocations that can be emitted from the crack tip along one slip plane as a function of grain size d.