1 Introduction

Interfaces can strengthen polycrystalline aggregates and multiphase composites because they act as strong barriers for slip transmission from one to the adjacent grain^[1-8]. In particular, interface properties play a crucial role in determining mechanical properties of multiple layers when the layer thickness is decreased to a few nanometers,because the dominating plastic mechanism is associated with dislocation transmission from one layer to the adjacent layer^[9-21]. Correspondingly,the concept of the weak-interface strengthening mechanism has been proposed and demonstrated in metallic multiple layers^[22-24].

When an interface has weak shear resistance,such as Cu/Nb metallic multiple layers^[25] and metal/amorphous interfaces^[26], lattice dislocations are trapped in interfaces and spread their cores by shearing the interface. In this paper,we aim at deducing the elastic fields associated with a dislocation with a spreading core at the interface of anisotropic bimaterials. The elastic fields associated with a dislocation with a condensed core have been studied for decades,including dislocation dynamics simulations^[27-33],the heterogeneous anisotropic elasticity theory^[34],the correspondence theorem of couple stress elasticity^[35],and the Stroh formalism^[36-39]. Here,we propose a conic model to mimic the distribution of the Burgers vector of a dislocation.

2 Modeling interface dislocation core-spreading

The actual distribution of a dislocation core depends on the interface properties,such as the interactions among interface atoms,crystal structures,crystal or grain orientations,and interface defects. One can get the actual dislocation core with the disregistry analysis of the dislocation at the interface from atomistic simulations. The distribution of the Burgers vector corresponds to the slope of the disregistry. For instance,Dholabha et al.^[40] obtained the disregistry analysis of the STO/MgO interface where an edge dislocation spreads at the interface. Wang et al.^[25] analyzed the disregistry of the interface atoms in Cu/Nb metallic multilayers. By taking the slope of their data of disregistry analyses,we find that (i) the slope is the largest at the center of the spreading region,which means that the Burgers vector has the maximum value at the center; (ii) the slope gradually decreases to zero from the center to the edge,which implies that the Burgers vector tends to zero at the spreading boundary; (iii) the whole slope curve is most likely smooth at the spreading region. Therefore,we propose a conic model to mimic the actual distribution of the dislocation core. This conic model can capture the fundamental characteristics and is illustrated in Fig. 1(a). Figure 1(b) is the corresponding disregistry plot.

For the conic model,the distribution of the Burgers vector can be expressed as

(1)

where $ b$ represents the distributed Burgers vector of an infinitesimal dislocation in the spreading region,and $ b_{0}$ is the summation of the Burgers vectors of the infinitesimal dislocation array. When the spreading dislocation is located at the interface ($Z=0$) with the dislocation line along the $y$-axis,for simplicity,the spreading region is assumed from $(-nb_{0},0)$ to $(nb_{0},0)$ in the $xOz$ coordinates,as shown in Fig. 2.

The relation between $ b$ and $ b_{0}$ can be expressed as

(2)

where the coordinate ($X$,$Z)$ is applied for the dislocation source,called the source point,while the coordinate ($x$,$z)$ is for other points or ordinary points,called field points. Substituting Eq.(1) into Eq.(2),we can get

(3)

Due to the conservation of the Burgers vectors,the sum of the Burgers vectors of the infinitesimal dislocation array should be a constant. Therefore,the only variable parameter in the previous conic mathematic model is the width of the core-spreading region $2nb_{0}$. The magnitude of the width depends on the properties of the interface,especially the interface shear strength. The smaller the interface shear strength,the wider the core-spreading region.

3 Elastic fields due to dislocation core spreading

Considering the nature of the core spreading (represented by distributed dislocations),the elastic fields of a spreading core dislocation can be obtained by superposing the elastic fields of each infinitesimal dislocation.

For a given dislocation with the compact core,the elastic fields have been studied by Barneet and Lothe^[41],Dundurs and Mura^[42],and Ting^[43]. Using the Stroh formalism^[44-45],the displacements of an interfacial dislocation with a compact core can be expressed.

For the source point ($X$,$Z)$ and the field point ($x$,$z)$ in Material 1,

(4)

with

For the source point ($X$,$Z)$ in Material 1 and the field point ($x$,$z)$ in Material 2,

(5)

For the source point ($X$,$Z)$ in Material 2 and the field point ($x$,$z)$ in Material 1,

(6)

For the source point ($X$,$Z)$ and the field point ($x$,$z)$ in Material 2,

(7)

where

and

In Eqs.(4)--(7),the superscript T denotes the transpose of the matrix,and the over-bar means complex conjugate. The superscripts and subscripts 1 and 2 denote Materials 1 and 2,respectively. The Stroh eigenvalues and the corresponding eigenmatrices are denoted by $p_j.{(\lambda )} $,$ A_\lambda $,and $ B_\lambda $ ($\lambda =1,$ 2). In Eqs.(4) and (7),the first term of the right-hand side corresponds to the full-space dislocation Green's function,and the second term is the image part (i.e.,the complementary part). This image part is caused by the interface or the inhomogeneity of the two half-spaces.

For a spreading core dislocation,we substitute Eq.(1) into Eqs.(4)--(7) and integrate the expression for the displacement directly with long and tedious procedure. The displacements can be written as

(8)

(9)

The first-order derivatives of the displacements in Eqs. (8) and (9) with respect to the field point x can be expressed as

(11)

(11)

The strain fields can be easily obtained according to Eqs.(10)--(11). Correspondingly,the stress fields can be derived through the constitutive~relation as follows.

4 Interfacial dislocation at Cu/Nb interface

We further examine the displacement and stress fields for an interfacial dislocation in Cu/Nb bimaterial. Due to the positive heat mixing between copper and niobium,the interfaces in Cu/Nb bimaterial are generally sharp^[46],as experimentally observed in CuNb multilayers that were fabricated by physical vapor deposition techniques. Atomistic simulations and experimental tests have demonstrated that the Cu/Nb interface with the Kurdjumov-Sachs (K-S) orientation has low shear resistance^{[19, 25]}. Molecular dynamic simulations have also provided insights into understanding the weak shear resistance of Cu/Nb interface and the role of weak shear interface in strengthening Cu/Nb composites. Here,we test the elastic fields in the Cu/Nb bimaterial. The upper half space of the bimaterial is copper,while the lower half space is niobium. The elastic moduli are $C_{11}=168.4$,GPa,$C_{12}=121.4$,GPa,and $C_{44}=75.4$,GPa for copper,and $C_{11}=246.0$,GPa, $C_{12}=134$,GPa,and $C_{44}=28.7$,GPa for niobium. The $x$-axis is parallel to ^[11-2]$_\mathrm {Cu}$ and ^[1-12]$_\mathrm {Nb}$,the $z$-axis is parallel to [111]$_\mathrm {Cu}$ and [110]$_\mathrm {Nb}$,and the $y$-axis points towards the paper and is parallel to [-110]$_\mathrm {Cu}$ and ^[1-1]$_\mathrm {Nb}$.

When a lattice dislocation enters the interface,the component of the Burgers vector parallel to the interface can easily spread out corresponding to the interface shear mechanisms. However,the component perpendicular to the interface plane is hard to spread out corresponding to the climb mechanisms^{[17, 47]}. The component parallel to the interface plane can be further decomposed into a screw part and an edge part. Therefore,the elastic fields of a dislocation can be obtained by superposing three dislocations,i.e., one compact dislocation,one screw dislocation with a spreading core,and one edge dislocation with a spreading core.

Considering the complexity in deriving and programming,the accuracy of the simulating results should be verified,which can be realized through the interface displacement and stress conditions. In the numerical simulation,the dislocation has the Burgers vector 1/2[101] on the glide plane (11$\bar{1}$). The width of the spreading region is set to be 10$b_{0}$ according to Wang et al.^[25]. For brevity,the compact core dislocation is denoted by Model 1,and the spreading core dislocation is denoted by Model 2. The displacement fields of the two models are shown in Figs. 3--4. Figure 3 reveals the following conclusions. (i) The displacement fields at the right hand of the core region are continuous,but there is a displacement jump at the left hand of the region. Because of the spreading region with a width $w$,the dislocation jump region of the conic model is larger than the compact model. (ii) In the core region,the displacement is not continuous across the interface in the conic model. The displacement jump coincides with the component of the Burgers vector in the $x$-direction,and the error between them is less than 0.1 %. (iii) The displacement fields at the field points far from the core region are almost the same,which means that the displacements are not sensitive to the distribution of the Burgers vector at the regions far from the dislocation core. We can obtain similar results as shown in Fig. 4. In conclusion,the displacement jump condition across the interface is verified.

The stress fields ($\sigma _{11}$,$\sigma _{13}$,$\sigma _{33})$ of the two models are shown in Figs. 5--7. The stress component $\sigma _{11}$ is not continuous across the interface ($z=0$),as shown in Fig. 5,which is ascribed to the heterogeneity property of the upper and lower layers. The stresses $\sigma _{13}$ and $\sigma _{33}$ across the interface are continuous,as shown in Figs. 6 and 7,which means that the interface stress continuity conditions are well satisfied. The stress fields induced by the condensed dislocation,as shown in Figs. 5(a),6(a),and 7(a),have severe stress concentration in the core region,while the stress concentration in the conic model is reduced greatly. The stress fields in the core region for the condensed model and conic model are different. The stress distribution of the conic model may be regarded as a kind of stretch (in the $x$-axis direction) of the stress concentration in the compact model with the core spreading and the stress decreasing. Overall,the magnitude of the stress fields decreases with the increasing distance from the dislocation.

In order to give more quantitative description,the stresses along two special axes (the $x$- and $z$-axes) are analyzed. Figure 8 shows the stress fields along the line $x=0$ for the compact model and conic model. The singularity in Model 1 is eliminated in the conic model. Besides,the stress components $\sigma _{13}$ and $\sigma _{33}$ in the conic model are continuous across the interface ($z=0$),while there exists a stress jump for $\sigma _{11}$,which is consistent with the observation in Figs. 6--7. Near the core region,the magnitude of the stress fields of Model 1 is much larger than that of Model 2,which implies the decrease of the stress concentration. Outside the core region,the stress fields decrease gradually with the increase of the distance from the center of the dislocation. Meanwhile,their difference becomes negligible.

Figure 9 shows the stress distribution near the interface. Since the stress component $\sigma _{11}$ is discontinuous across the interface,we cannot simply assign the coordinate $z=0$. The stress fields in Fig. 9 refer to $z=-0.002 b_{0}$. Similar conclusions can be obtained as from Fig. 8.

As stated previously,the magnitude of the stress fields decreases with the increasing distance of the field point to dislocation position in the contour plot,but less quantitative analysis has been reported. For the purpose of engineering applications,more attention should be paid to the difference between the spreading core model and the compact core model. Since the maximum shear stress often plays a critical role in plastic deformation of metallic materials,it is selected as the standard to evaluate the influence of the two different models. Figure 10 shows the relation between the maximum shear stress and the distance from the dislocation,where the maximum shear stress denotes the largest shear stress along the circle with a radius $r$ from the dislocation center. With the increase of the distance $r$, the difference between the maximum shear stresses of the two models decreases. The relative error of the maximum shear stresses is defined as

(12)

where $\tau _{\max }.1 (r)$ and $\tau _{\max }.2 ( r )$ denote the maximum shear stresses in Model 1 and Model 2 along the circle with the radius $r$,respectively. From Fig. 10,we can find that the relative error is large near the core region,while it is small at the region far away from the dislocation. Moreover,in Model 2,the relative error is lower than 5 % when $r>1.5w$,which means that the difference between the two models can be negligible when $r$ is greater than 1.5$w$.

5 Conclusions

Using the Stroh formalism and Green's function,we deduce the analytical expressions of the elastic fields for interfacial dislocations that spread their core within the interface in anisotropic bimaterials. We further examine the fields in Cu/Nb bimaterial and find the following conclusions. (i) The accuracy and efficiency of the method are validated by the interface conditions. (ii) A spreading core can greatly reduce the stress intensity near the interfacial dislocation compared with the compact core. (iii) The elastic fields near the spreading core region are significantly different from the condensed core,while they are less sensitive for a field point which is 1.5 times the core width away from the center of the spreading core. The results of this work are important for multiple layer composites,especially for layer thickness on the nanometer level,because the layer thickness is comparable with the size of the spreading core. We expect that the results from this work can be applied to design of nanoscale multilayers^[48-49].