高阶应变梯度弹性理论的新型微分求积单元法
Novel differential quadrature element method for higher order strain gradient elasticity theory
摘要：
提出了一种基于Lagrange插值的新型微分求积单元，用于求解非经典梁理论中的六阶偏微分方程。这些非经典理论将位移、斜率和曲率作为欧拉-伯努利梁的自由度。本文提出了一种通用的方案，以简化和有效的方式实现与这些非经典理论相关联的多自由度自由度。建议的元素有位移作为唯一的自由度在域中，而在边界上，它有位移，斜率和曲率。此外，我们扩展了这种方法，并制定了两个新版本的梯度弹性理论的板元。在第一个版本中，拉格朗日插值假定在$x$和$y$方向和第二个版本是基于混合插值，与拉格朗日插值在$x$方向和Hermite插值在$y$方向。通过结合经典和非经典边界条件计算修改后的加权系数的程序进行说明。通过对梯度弹性梁和板在不同边界条件下的静态分析的数值算例，验证了所提出单元的有效性。
英文摘要：
In this paper, we propose a novel and efficient differential quadrature element based on Lagrange interpolation to solve a sixth order partial differential equations encountered in non-classical beam theories. These non-classical theories render displacement, slope and curvature as degrees of freedom for an Euler-Bernoulli beam. A generalize scheme is presented herein to implementation the multi-degrees degrees of freedom associated with these non-classical theories in a simplified and efficient way. The proposed element has displacement as the only degree of freedom in the domain, whereas, at the boundaries it has displacement, slope and curvature. Further, we extend this methodology and formulate two novel versions of plate element for gradient elasticity theory. In the first version, Lagrange interpolation is assumed in $x$ and $y$ directions and the second version is based on mixed interpolation, with Lagrange interpolation in $x$ direction and Hermite interpolation in $y$ direction. The procedure to compute the modified weighting coefficients by incorporating the classical and non-classical boundary conditions is explained. The efficiency of the proposed elements is demonstrated through numerical examples on static analysis of gradient elastic beams and plates for different boundary conditions.
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