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 引用本文: 苏文政, 刘书田. 一类多孔固体的等效偶应力动力学梁模型[J]. 力学学报, 2016, 48(1): 111-126.
Su Wenzheng, Liu Shutiany. EFFECTIVE COUPLE-STRESS DYNAMIC BEAM MODEL OF A TYPICAL CELLULAR SOLID[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(1): 111-126.
 Citation: Su Wenzheng, Liu Shutiany. EFFECTIVE COUPLE-STRESS DYNAMIC BEAM MODEL OF A TYPICAL CELLULAR SOLID[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(1): 111-126.

## EFFECTIVE COUPLE-STRESS DYNAMIC BEAM MODEL OF A TYPICAL CELLULAR SOLID

• 摘要: 一维多孔固体结构可采用等效连续介质梁模型来研究其动力学行为. 当类梁结构的高度尺寸和多孔固体单胞结构尺寸相近时,等效模型的力学行为会产生尺寸效应现象. 等效经典模型由于不包含尺度参数而无法描述尺寸相关特点,而广义连续介质力学模型则可以准确地考虑尺寸效应的影响. 基于偶应力理论,对一类单胞含有圆形孔洞的周期性多孔固体类梁结构,给出了分析其横向自由振动的等效连续介质铁木辛柯梁模型. 通过对单胞分析,在应变能等价和几何平均的意义下,定义了等效偶应力介质的材料常数. 利用已有的材料常数,推导了等效铁木辛柯梁的动力学微分方程. 将实际多孔固体结构进行完全的动力学有限元离散计算,所获得的解作为精确解以检验等效梁模型所获得的频率和振型的精度. 振型的比较借助于模态置信准则矩阵方法. 大量算例表明,等效偶应力铁木辛柯梁模型在频率和振型两方面均具有较高的计算精度. 重点研究了单胞孔径的相对大小、类梁结构高度与单胞尺寸比以及类梁结构长高比对等效梁模型精度的影响. 在此基础上,偏保守地建议了多孔固体类梁结构自振分析方法.

Abstract: The e ective continuum beam model can be used to study the dynamic behavior of the one-dimensional cellular solid structure. However, this e ective model will bring the size e ect when the cell size is close to the height dimension of the beam-like structure. The classical continuum model, which has no length scale, cannot describe this size dependent feature. By contrast, the generalized continuum model is pertinent to the description of the size e ect. The purpose of this paper is to develop an e ective Timoshenko continuum beam model for the transverse natural vibration analysis of a periodic cellular solid beam-like structure whose unit cell has a circular void, based on the couple-stress theory. The properties of the e ective couple-stress continuum are generated under the criterion of the equivalent strain energy and the geometrical average by the analysis of a unit cell. These properties are then employed in the Timoshenko beam theory to obtain the dynamics di erential equations. Comparison is made with the prediction of a finite element analysis of the complete cellular structure, which is taken as the benchmark for accuracy. Good agreements both on the frequencies and mode shapes are found by a variety of examples where the method of modal assurance criterion (MAC) is employed for the comparison of the mode shapes. The emphasis is put on the influence of the relative size of the void diameter, the size ratio of the beam height to the cell size, and the aspect ratio of the beam-like structure on the accuracy of the e ective beam model. A series of methods on the natural vibration analysis of a cellular solid beam-like structure is recommended conservatively based on these analyses.

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