Peierls-Nabarro model of interfacial misfit dislocation: An analytic solution

We propose a method to treat the interfacial misfit dislocation array following the original Peierls-Nabarro's ideas. A simple and exact analytic solution is derived in the extended Peierls-Nabarro's model, and this solution reflects the core structure and the energy of misfit dislocation, which depend on misfit and bond strength. We also find that only with beta < 0.2 the structure of interface can be represented by an array of singular Volterra dislocations, which conforms to those of atomic simulation. Interfacial energy and adhesive work can be estimated by inputting ab initio calculation data into the model, and this shows the method can provide a correlation between the ab initio calculations and elastic continuum theory.

The modified Peierls-Nabarro model of interfacial misfit dislocation

The Peierls-Nabarro model of the interfacial misfit dislocation array is analytically extended to a family of dislocations of greater widths. By adjusting a parameter, the width of the misfit dislocations, the distribution of the shear stress, and the restoring force law can be systematically varied. The smaller the amplitude of the restoring force, the wider the misfit dislocations and the lower the interfacial energy.

A new method to analyse the Peierls-Nabarro model of an edge dislocation in a rectangular plate

A new method is presented here to analyse the Peierls-Nabarro model of an edge dislocation in a rectangular plate. The analysis is based on the superposition scheme and series expansions of complex potentials. The stress field and dislocation density field on the slip plane can be expressed as the first and the second Chebyshev polynomial series respectively. Two sets of governing equations are obtained on the slip plane and outer boundary of the rectangular plate respectively. Three numerical methods are used to solve the governing equations.

基于跨尺度模型的界面研究

探索和建立不同尺度理论之间的关联模式是科学研究的重要课题，本文基于跨尺度模型着重探讨了金属陶瓷界面的凝聚能和原子结构问题。本文遵循原始Peierls-Nabarro模型的基本思想，提出了一种处理一维界面失配位错组的新方法。在这个推广的Peierls-Nabarro模型中，本文得到了一个简单而且准确的解析解，此解反映了失配位错的核结构、能量与失配度、剪切模量之间的依赖关系。当界面剪切模量较强而失配度较小时，界面的结构可以用一组奇导师Volterra位错来描述，这与一些原子模拟结果一致。采用这一简单的模型，引入第一原理计算得到的数据，此模型可以估算金属陶瓷界面的凝聚能。一维界面失配位错组的Peierls-Nabarro模型还被解析推广描述一大类较宽的位错。在模型中我们引进了一个参数a，通过控制参数a，我们可以系统地改变失配位错芯的宽度、剪切应力的分布和弹性恢复力。随着a增加，位错宽度增加，同时弹性恢复力和失配位错应力的幅度减少。当界面剪切模量强和失配度小时，失配位错的宽度近似线性反比于弹性恢复力的幅度大小。同时当界面剪切模量和失配度固定时，失配能、弹性能和总的界面能随a的增加而减少。界面能和恢复力律形式密切相关，当界面剪切模量弱和失配度大时，这种依赖关系更强。考虑到界面常常是在晶格两个方向都有失配，本文还引进了描述界面周期失配位错的二维广义Peierls-Nabarro模型，使得我们能够定量地研究界面的结构和能量。文中定量分析了广义堆垛能γ面对界面失配位错的结构和能量的影响，分析了位错网中两种位错组的相互作用。当界面剪切模量τ_0变大和失配度f变小时，随着位错核区占整个界面的比重下降，γ面的形状对界面能量和结构影响减弱，结果两种位错组之间的相互作用也减弱。此外γ面的变化还有可能导致位错网结构的转变，也就是导致界面结构的转变。应用此模型，本文还研究了金属－陶瓷Ag/MgO(100)界面，给出了界面的能量和原子结构。文中得出结论：在Ag/MgO(100)界面将会形成{1/2<110>; <110>}类型的位错网。此外由于界面失配位错的形成，Ag/MgO(100)界面凝聚能的理论值900mJ/m~2将减少214mJ/m~2，最终成为686mJ/m~2。基于第一原理赝势平面波的总能计算，文中给出了金属陶瓷Al/MgO(100)界面弛豫和未弛豫时的广义堆垛能面。然后结合第三章发展的广义二维Peierls-Nabarro模型，详细研究了金属陶瓷Al/MgO(100)界面的原子结构和界面能。文中得出的“在Al/MgO(100)界面将会形成{1/2<110>; <110>}类型位错网”的推论，证实了Vellinga等的猜测；文中还预测了凝聚能的理论是在600mJ/m~2（未弛豫情形）和670mJ/m~2（弛豫情形）之间。这个应用表明此方法能够容易地建立连续介质理论和第一原理计算之间的联系，实现理论上的跨尺度。本文最后提出了一种得到界面原子有效对势的反演方法。通过反演金属－MgO陶瓷界面的第一原理计算的凝聚能曲线，我们得到了一些金属原子和陶瓷离子之间的对势，此对势反映了金属陶瓷键合的特性。本文的反演方法提供了通过第一原理计算数据来拟合界面原子对势的一种可行性途径。这种方法可归结为第一类尺度关联理论，即单向的跨尺度关联模式。

A unified model for dislocation nucleation, dislocation emission and dislocation free zone

In this paper, a unified model for dislocation nucleation, emission and dislocation free zone is proposed based on the Peierls framework. Three regions are identified ahead of the crack tip. The emitted dislocations, located away from the crack tip in the form of an inverse pileup, define the plastic zone. Between that zone and the cohesive zone immediately ahead of the crack tip, there is a dislocation free zone. With the stress field and the dislocation density field in the cohesive zone and plastic zone being, respectively, expressed in the first and second Chebyshev polynomial series, and the opening and slip displacements in trigonometric series, a set of nonlinear algebraic equations can be obtained and solved with the Newton-Raphson Method. The results of calculations for pure shearing and combined tension and shear loading after dislocation emission are given in detail. An approximate treatment of the dynamic effects of the dislocation emission is also developed in this paper, and the calculation results are in good agreement with those of molecular dynamics simulations.