Boundary Value Problems of a Kirchhoff Type Plate Model Based on the Simplified Strain Gradient Elasticity and the Application
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摘要:
考虑应变梯度和速度梯度的影响，建立薄板控制微分方程及给出其边值问题的提法，修正了前人给出的薄板角点条件。采用Levy法，给出受分布力作用下简支板的挠度及自由振动频率的解析解。通过与文献中分子动力学数据对比，验证了该文模型的有效性并提出校核材料参数的一种方法。研究结果表明，增大弹性地基和应变梯度参数可以有效提高板的等效刚度，而速度梯度参数则相反。该文提出的板的边值问题为研究薄板在复杂支撑边界及外荷载等条件提供了理论依据。同时，有望为其有限元法、有限差分法和基于能量原理的Galerkin法等数值方法提供理论依据。
Abstract:A new type of thin plate model and the related nonclassical boundary value problems were established within the framework of strain gradient and velocity gradient elasticity. The closed-form solutions of deflections and free vibrational frequencies of a simply supported plate resting on an elastic foundation were obtained. The results of the present model agree well with those predicted by the molecular dynamics. Numerical results show that, the elastic foundation and the strain gradient parameter have a stiffness-hardening effect, while the velocity gradient parameter has a stiffness-softening effect. The proposed boundary value problems are of great significance to the study of the mechanical behaviors of plates under complex boundary conditions and external loadings. Furthermore, it will be useful for developing effective numerical methods such as the finite element method, the finite difference method and the Garlerkin method.
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Key words:
- plate /
- vibration /
- deflection /
- strain gradient theory /
- variational principle
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图 2 弹性地基上的周边简支矩形板
Figure 2. A fully simply supported rectangular plate resting on an elastic foundation
图 3 周边简支方形板的基频与其边长的关系
Figure 3. The fundamental frequency vs. the side length of a simply supported square plate
图 4 地基刚度系数对周边简支方板位移形状的影响
Figure 4. Effects of the foundation stiffness on the displacement of a simply supported square plate for y=a/2
图 5 地基刚度系数对周边简支方板基频的影响
Figure 5. Effects of the foundation stiffness on the fundamental frequency of a simply supported square plate
图 6 应变梯度参数l2对周边简支方板位移形状的影响
Figure 6. Effects of strain gradient parameter l2 on the displacement of a simply supported square plate for y=a/2
图 7 应变梯度参数l2对周边简支方板基频的影响
Figure 7. Effects of strain gradient parameter l2 on the fundamental frequency of a simply supported square plate
图 8 速度梯度参数l1对周边简支方板位移形状的影响
Figure 8. Effects of velocity gradient parameter l1 on the displacement of a simply supported square plate for y=a/2
图 9 速度应变梯度参数l1对周边简支方板基频的影响
Figure 9. Effects of velocity gradient parameter l1 on the fundamental frequency of a simply supported square plate
A1 坐标系(x, y)和(n, s)及板边界Γ
A1. Coordinate systems (x, y) and (n, s) at a piecewise smooth plate boundary Γ
表 1 矩形薄板3种常见的边界条件
Table 1. Three common boundary conditions (BCs) for a rectangular plate
boundary condition BC1 BC2 BC3 clamped w=0 w,x=0 Mxxx=0 or w,xx=0 simply support w=0 M*xx=0 Mxxx=0 or w,xx=0 free $ Q_x^* + \rho hl_1^2{\ddot w_{,x}} = 0 $ M*xx=0 Mxxx=0 or w,xx=0 
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