In gradient enhanced crystal plasticity formulations, additional conditions are required on grain boundaries which impact slip system activity in their neighborhood. In the case when gradient effects are caught through a dislocation density tensor, these conditions are prescribed on the flow of this tensor through the boundary or on the dual microforce. Gurtin (2008) proposed to derive the grain boundary behavior from a free energy potential function of a surface dislocation density tensor, the evolution of which equals the balance of flows on both sides. In this case, the response of a bi-crystal is driven by energy minimizing considerations extending the approach developed by Petryk and co-workers to gradient plasticity and grain boundary (GB) behavior: slip activity is then an optimum between deformation accommodation and limitation of accumulated bulk and GB defects energies and depends on their respective moduli. This is illustrated for the one dimensional problem of a bi-layered strip in simple shear and different cases are analyzed in terms of slip system orientations and strain hardening. Still, in the case of low angle GBs, the internal structure is assumed to be constituted of dislocation arrays accommodating the lattice misorientation between the two grains; an intrinsic dislocation density tensor can be associated to this structure and incorporated in the GB modeling.

在梯度增强的晶体可塑性配方中，在晶界上需要附加条件，这会影响滑移系统在其附近的活性。在通过位错密度张量捕获梯度效应的情况下，这些条件是在该张量流过边界或双重微力时规定的。Gurtin（2008）提出从表面位错密度张量的自由能势函数得出晶界行为，其演化等于两侧流动的平衡。在这种情况下，双晶的响应受到能量最小化考虑因素的驱动，从而将Petryk及其同事开发的方法扩展到梯度可塑性和晶界（GB）行为：因此，滑动活动性在变形调节与累积体积和GB缺陷能量的限制之间是最佳的，并取决于它们各自的模量。对于简单剪切下的双层带材的一维问题，可以说明这一点，并根据滑移系统的方向和应变硬化分析了不同的情况。此外，在低角度GBs的情况下，内部结构仍假定为由能容纳两个晶粒之间晶格取向失调的位错阵列构成。固有位错密度张量可以与此结构相关联，并纳入GB建模中。这是针对简单剪切中的双层带材的一维问题而说明的，并且根据滑动系统的方向和应变硬化分析了不同的情况。此外，在低角度GBs的情况下，内部结构仍假定为由能容纳两个晶粒之间晶格取向失调的位错阵列构成。固有位错密度张量可以与此结构相关联，并纳入GB建模中。对于简单剪切下的双层带材的一维问题，可以说明这一点，并根据滑移系统的方向和应变硬化分析了不同的情况。此外，在低角度GBs的情况下，内部结构仍假定为由能容纳两个晶粒之间晶格取向失调的位错阵列构成。固有位错密度张量可以与此结构相关联，并纳入GB建模中。