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基于一种新修正偶应力理论建立了微尺度平面正交各向异性功能梯度梁的自由振动模型。模型中包含两个材料尺度参数,能够分别描述两个正交方向上不同程度的尺度效应。当梁的几何尺寸远大于材料尺度参数时,本文模型亦可自动退化为相应的传统宏观模型。基于哈密顿原理推导了运动控制方程并以简支梁的自由振动为例分析了几何尺寸、功能梯度变化指数等对尺度效应产生的影响。算例结果表明:采用本文模型所预测的梁自振频率总是大于传统理论的结果,即捕捉到了尺度效应。尺度效应会随着梁几何尺寸的增大而逐渐减弱并在几何尺寸远大于尺度参数时消失;高阶自振频率所体现出的尺度效应较低阶自振频率更加明显。此外,功能梯度变化指数对尺度效应也有一定的影响。
A free-vibration model of a microscale planar orthotropic functional gradient beam is established based on a new theory of even-induced stress. The model contains two material-scale parameters that describe the different scale effects in two orthogonal directions, respectively. When the beam geometry is much larger than the material scale parameters, the model can be automatically degenerated into the corresponding traditional macroscopic model. Based on the Hamilton principle, the equations of motion control are deduced. Taking the free vibration of simply supported beams as an example, the influence of geometric dimension and function gradient index on the scale effect is analyzed. The results of the example show that the natural frequency of the beam predicted by this model is always larger than that of the traditional theory, that is, the scale effect is captured. The scale effect will be weakened with the increase of the geometric dimension of the beam and disappear when the geometric dimension is much larger than the scale parameter. The higher the natural frequency, the lower the natural frequency is, the lower the natural frequency is. In addition, the function gradient index also has some effect on the scale effect.