Concurrent interface shearing and dislocation core change on the glide dislocation-interface interactions: a phase field approach

Strengthening in nanoscale metallic multilayers is closely related to the glide dislocation-interface interaction. The interface can be sheared by the stress of the approaching glide dislocation with its core changed. How the concurrent interface shearing and the dislocation core change influence such interaction dominated strength is studied using three dimensional phase field microelasticity modeling and simulation. The simulated results show that when the glide dislocation is close to or away from the interface, the width of its core changes abruptly in accompany with the interface shear zone broadening or shrinking, respectively. A wider interface shear zone is developed on the interface with a lower shear strength, and can trap the glide dislocation at the interface in a lower energy state, and thus leads a stronger barrier to dislocation transmission. The results further show that the continuum model of the dislocation without the core-width change underestimates the interfacial barrier strength especially for the glide dislocation transmission across weak interfaces.


Introduction
Computer modeling and simulation of defects ensemble and their elastic interactions from atomistic scale to continuum scale are important to get insights into mechanism-based strength or plasticity in materials [1,2]. Especially computational modeling capable of bridging multiple time and length scales becomes one of the fast growing areas in understanding the mechanical response of materials [3]. In metallic multilayers, the strength shows a maximum at a critical layer thickness [4][5][6][7]. The maximum is closely related to the critical stress required to transmit a glide processes and the interface shearing process can be addressed as well [36]. In this paper such phase field approach is adopted to discuss the effect of concurrent interface shearing and dislocation core change on the interfacial barrier strength for the glide dislocation transmission. It is worth to mention that recently the large-deformation phase field theory is developed and applied for the interaction between dislocations and phase interfaces [38][39][40][41][42][43][44]. The important mechanism of interface motion elucidated by this improved model is enlightening for dislocation-interface interaction. Although in the current study we use the small-deformation phase field model, and these calculations are not yet material specific because the simple stacking fault energy surfaces for the glide plane and the interface are adopted. It would provide valuable trend in what determines the width and depth of the energy well, the interfacial barrier strength for slip transmission and give insight into the limitations of continuum model of the dislocation without the core-width change.

Phase Field Model of the Glide Dislocation Across a Coherent Sliding Interface
In the current model we consider a screw dislocation across a bi-material coherent interface shown in Figure 1. The glide plane is denoted by 0 z  and the interface between phases I and II is viewed as a mathematically sharp plane denoted by 0 x  . To focus on the effects of the concurrent interface shearing and the dislocation core change, the lattice mismatch between the two phases is ignored. A pre-existing dislocation at the glide plane in phase II tends to glide across the interface under an external shear, ext yz  . The straight glide dislocation with the Burgers vector and the line direction both parallel to y axis is assumed to be a pure screw type for simplicity, the resultant shear The gradient term included in the original PFM models [27] is removed here for simplicity followed by the previous work [32,33,36].
Here two phase field variables, are in the unslipped area at the glide plane and the interface respectively. Following the treatment in the PFM, formulating the total free energy in Eq.(1) as a function of the two phase fields can be obtained. The crystalline energy (interplanar potential energy) caused by localized slip is given by where  is the density of interplanar potential energy as a functional of the interplanar slip  (5) and at the interface between phase I and II, int Δ is given by br bb (6) where the Burgers vector, the interplanar distance and the unstable SFE are s  for the interface, respectively. (2) reduces to that of the conventional crystalline energy function in terms of the inelastic slip [27]. Otherwise, the effect of the elastic slip cannot be ignored in particular during dislocation motion under an applied stress [44]. The elastic energy in Eq. (1) is mainly caused by the interplay between the external stress and the eigenstrain distribution from the inelastic slips. The eigenstrain where s j n and int j n are the component of the unit normal vector of the glide plane and of the interface respectively. When the bi-material system is elastically homogeneous, the Khachaturyan-Shatalov theory [45,46] gives the exact solution of the elastic energy in terms of where ijkl C is the elastic moduli tensor, ξ is the vector in the Fourier space, and /  e ξ ξ is the After the total free energy is expressed as a function of the given two phase field variables, the , r rr (10) where t is the time and L is the positive kinetic coefficient characterizing the relaxation rates of

Results and Discussion
The interfacial barrier to glide dislocation slip transmission is influenced by both the dislocation core changes and the cross-slip interaction between the glide dislocation and induced interfacial dislocations. The results consist of two sub-sections. In Section 3.1, the interface is assumed to be strongly bonded so that it is non-shearable, and the effect of the SFE mismatch on the dislocation core change as well as the interface resistance is investigated. In Section 3.2, the interface with weak interfacial bonding can be sheared by the stress field of the glide dislocation and the influence of the interface shear on the interfacial barrier is studied.

The interfacial resistance of a non-shearable interface
If the interface is non-shearable, int  is viewed as infinite and we assumed  The rapid decrease of the core width near the interface would cause large energy change of the system during the transmission process. dislocation core change due to the SFE mismatch between the two phases. Figure 4 plots   According to the definition of the interfacial barrier strength in Figure 4, the interfacial barrier strengths without interface shear are calculated for different I  and II  . Figure 5 The calculated result is consistent with the analytic solution given by Anderson and Xin [14] in the case of no elastic mismatch. These results demonstrate that the larger SFE mismatch induces the larger dislocation core change during the transmission process and causes more rapid change of the dislocation energy near the interface, and leads to larger interfacial resistance.

Influence of the interface shear on the interfacial resistance
When the interface has limited shear strength and it could be sheared by the stress field of the glide dislocation. To investigate the effects of interface shear, we keep I . In comparison with * 0   in the similar case without interface shear as simulated in Figure 5, we believe that it is the interface shear leading to the significant increase of *  , which is consistent with the previous results of atomistic simulation and modeling [21][22][23][24][25]. From Figure 7, the dislocation core is constricted heavily when it reaches the interface. The reason is that the interface shear can reduce the elastic energy by counteracting the shear stress, the dislocation core tends to be trapped and constricted onto the sharp interface to relieve the elastic energy.
The interface shearing at the interface results in an inelastic shear zone via nucleation and growth of interfacial dislocations.  r during the glide dislocation transmission is shown in Figure 9.
The snapshots in Figure 9 clearly show that the inelastic shear zone adaptively changes through the reversible movement of the interfacial dislocation driven by the stress field of the glide dislocation close to or away from the interface.    . It is found that the interfacial barrier strength significantly increases as the value of int  is lower than 0.8. When the value of int  is larger than 0.8, the interfacial barrier strength reaches a minimum (zero), which is the same as the prediction in the case without interface shear. Herein the enhancement of the interfacial barrier strength due to the interface shearing is neglectable. This is because the inelastic shear zone almost disappears at int 0.8

 
as indicated in Figure 10. These results reproduce some atomistic simulations trends revealing the "weak" interface strengthening mechanism [25]. The resistant force of the glide dislocation with interface shearing is originated from the increase of the total free energy as the glide dislocation away from the interface. Figure 12  . The deeper the energy well at the interface, the larger the attractive force on the glide dislocation, and the higher the interfacial barrier strength. The results in Figure 12 indicate that the attractive force is attributed to not only the elastic  interaction between the glide dislocation and interfacial dislocation [12,18] but also the crystalline energy change due to the core-width change of the dislocations [16]. At the continuum scale, the dislocation core change is usually ignored in the dislocation-based models [12,18]. In such situation the resistant force on the glide dislocation is only related to the change of the elastic energy and therefore is underestimated. The current simulation results show that the change of the dislocation crystalline energy induced by the core change also has large contribution to the interfacial barrier strength, especially for the weak interface.

Conclusions
In summary, 3D PFM modeling and simulation are adopted to investigate a glide dislocation transmission across a coherent sliding interface. The interfacial barrier strength for the transmission is investigated with and without interface shear. In the case without interface shear, the resistant force on the glide dislocation across the interface mainly depends on the core-width change induced by the SFE mismatch, which is in good agreement with the theoretical solutions. In the case with interface shear, we show that weak interface develops a wide inelastic shear zone under the stress field of the glide dislocation, and can exert a large attractive force on the glide dislocation and thus largely enhance the interfacial barrier strength. The attractive force is attributed to both the elastic interaction between the glide dislocation and interfacial dislocation and the core change of them. The continuum model for the dislocation transmission across the weak interface without the core-width change may significantly underestimate the interfacial barrier strength.