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应变梯度Mindlin板边值问题的讨论

DISCUSSION ON BOUNDARY VALUE PROBLEMS OF A MINDLIN PLATE BASED ON THE SIMPLIFIED STRAIN GRADIENT ELASTICITY

  • 摘要: 实验和分子动力学计算结果表明, 当材料/结构的特征尺寸降为微纳米量级时, 他们将表现出明显的尺度效应, 因此能否建立精确表征其力学行为的连续介质力学模型具有重要的理论和现实意义. 尽管现有文献对非经典Mindlin板的力学行为进行了大量研究, 但该模型的变分自洽的边值问题是近年来未攻克的科学问题之一. 基于简化的应变梯度理论给出了各向同性Mindlin板应变能的表达式, 通过变分原理和张量分析, 得到了Mindlin板变分自洽的边值问题及其对应角点条件的位移微分表达式. 本文Mindlin板模型的边值问题可退化为相应的Timoshenko梁和Kirchhoff板模型的边值问题, 验证了本文结果的有效性. 研究结果发现, 该Mindlin板模型的控制方程是一个解耦后横向振动具有12阶的偏微分方程, 因此需要每个板边提供6个边界条件. 角点条件由双应力(double stress)产生, 并与经典的剪力、弯矩和扭矩沿截面的法向梯度有关. 本文首次澄清了应变梯度Mindlin板存在角点条件这一事实, 所得的变分结果有望为其有限元法和伽辽金法等数值方法提供理论依据.

     

    Abstract: It is reported from both the experiments and molecular dynamics that the materials and structures will exhibit a remarkable size-effect when their characteristic sizes shrink down to the micro- to nano-scales. Therefore, it is a hot topic of current interest in the research communities that whether is it possible to establish an accurate continuum model that is capable of predicting the mechanical behaviors of materials and structures. Although numerous studies have been carried out on the mechanical behaviors of Mindlin plates, their variationally consistent boundary value problems and the related issues have not been well addressed in the open literature. Firstly, the strain energy of an isotropic Mindlin plate within the context of the simplified strain gradient elasticity is given. Then, the variationally consistent boundary value problems of the Mindlin plate model and the corresponding corner conditions in terms of displacement derivations are derived using the variational principle and tensor analysis. It is verified that the present boundary value problems of the Mindlin plate model recover to the corresponding boundary value problems of the Timoshenko beam model and Kirchhoff plate model. It is found that the governing equation of the transverse behavior of the present Mindlin plate model involves a set of a decoupled twelfth-order partial differential equation, and therefore, should enforce six boundary conditions on each plate side, to constitute a well-posed boundary value problem. The possible boundary conditions of a circular plate and a rectangular plate are discussed. The corner conditions, produced by the double stresses, are closely related to normal gradients of classical components of the shear force, the bending moment and the twisting moment. The corner conditions, existed in the present Mindlin plate model, are firstly clarified. The present work is expected to be a useful tool for developing effective numerical methods, such as the finite element method and the Garlerkin method.

     

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