Two mixed meshless collocation methods for solving problems by considering a linear gradient elasticity theory of the Helmholtz type are proposed. The solution process is facilitated by employing operator-split procedures, splitting the original 4th-order problem into two uncoupled second-order sub-problems, which are then solved in a staggered manner by applying the mixed meshless local Petrov–Galerkin collocation strategy. Thereby, identical nodal pattern is used for the discretization of both lower-order problems, while the approximation of all unknown field variables is performed by applying the Moving Least Squares functions with interpolatory conditions. The performance of the derived methods is tested by some suitable numerical examples, dealing with elasticity problems that are often treated by gradient theories. Therein, it is demonstrated that both proposed methods are able to capture the size effect and that the strain-based method is able to produce the non-singular strain field around the crack tip, similar to the nonlocal elasticity approach of Eringen. It has been found out that the obtained results agree well with available analytical and numerical solutions.

中文翻译：

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亥姆霍兹型梯度弹性理论的混合无网格局部 Petrov-Galerkin (MLPG) 配置方法

提出了两种考虑亥姆霍兹型线性梯度弹性理论解决问题的混合无网格配置方法。求解过程通过采用算子拆分程序来促进，将原始四阶问题拆分为两个未耦合的二阶子问题，然后通过应用混合无网格局部 Petrov-Galerkin 搭配策略以交错的方式求解。因此，相同的节点模式用于两个低阶问题的离散化，而所有未知场变量的近似是通过应用具有插值条件的移动最小二乘函数来执行的。通过一些合适的数值例子来测试导出方法的性能，处理梯度理论经常处理的弹性问题。其中，结果表明，所提出的两种方法都能够捕获尺寸效应，并且基于应变的方法能够在裂纹尖端周围产生非奇异应变场，类似于 Eringen 的非局部弹性方法。已经发现，获得的结果与可用的解析解和数值解非常吻合。