This study presents a peridynamic (PD) correspondence model with strain gradient elasticity (SGE) to capture size effect on strength of nano- and micro-scale structures. The classical elasticity theory lacking a length scale parameter does not account for the microstructure-dependent size effects. Strain Gradient (SG) elasticity is an extension of the classical elasticity with a length scale parameter(s) which can be linked to the microstructure of the material. Peridynamic theory introduces damage into the constitutive relations in a natural way by allowing for interactions of a material point within its horizon which is referred to as the internal length parameter. The correspondence model of the PD theory permits the incorporation of the constitutive relationship of SG theory. Therefore, the present approach combines the effects of PD and SGE length scale parameters on the stiffness and strength of the material. The resulting equations present two length parameters: the horizon of a material point in PD theory and the characteristic length in SGE theory. The PD equation of motion with SGE (PDSG) is free of spatial derivatives and allows for the imposition of classical and nonclassical boundary conditions. The PDSG is first applied to investigate the deformation response of a one-dimensional nanoscale film subjected to a quasi-static tensile load and a dynamic load through an initial constant strain. The static response is compared with the analytical solution for varying ratio of SG length parameter to PD horizon. The dynamic response is compared with a computational solution while satisfying the nonclassical boundary conditions. Subsequently, the PDSG is applied to study the deformation response of a two-dimensional nanoscale film subjected to a quasi-static axial load and a specified tangential end displacement. These two cases are considered under special displacement constraints to compare with the available analytical solutions. Subsequently, a two-dimensional nanoscale film with or without an internal crack is subjected to uniform tensile loading. Finally, crack propagation is simulated under incremental displacement loading. The results capture the analytical predictions and the expected increase in stiffness with increasing ratio of SG length parameter to PD horizon.

中文翻译：

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具有应变梯度弹性的近场动力学对应模型，用于微观结构相关的尺寸效应

本研究提出了一种具有应变梯度弹性（SGE）的近场动力学（PD）对应模型，以捕获尺寸对纳米和微米尺度结构强度的影响。缺乏长度尺度参数的经典弹性理论无法解释与微观结构相关的尺寸效应。应变梯度 (SG) 弹性是经典弹性的延伸，其长度尺度参数可与材料的微观结构相关联。近场动力学理论通过允许质点在其视界内的相互作用（称为内部长度参数），以自然的方式将损伤引入本构关系。 PD理论的对应模型允许结合SG理论的本构关系。因此，本方法结合了 PD 和 SGE 长度尺度参数对材料刚度和强度的影响。所得方程提供了两个长度参数：PD 理论中的质点视界和 SGE 理论中的特征长度。 SGE 的 PD 运动方程 (PDSG) 没有空间导数，并且允许施加经典和非经典边界条件。 PDSG 首先用于研究一维纳米级薄膜在准静态拉伸载荷和初始恒定应变动态载荷作用下的变形响应。将静态响应与变化 SG 长度参数与 PD 层位之比的解析解进行比较。在满足非经典边界条件的情况下，将动态响应与计算解进行比较。随后，应用PDSG研究二维纳米级薄膜在准静态轴向载荷和指定切向末端位移作用下的变形响应。这两种情况在特殊位移约束下考虑，以与可用的解析解进行比较。随后，对有或没有内部裂纹的二维纳米级薄膜施加均匀的拉伸载荷。最后，模拟增量位移载荷下的裂纹扩展。结果捕获了分析预测以及随着 SG 长度参数与 PD 水平比的增加而预期的刚度增加。