基于偶应力理论的微结构振动分析无网格方法
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摘要
近年来,随着现代科技的迅猛发展,工业元器件日趋微型化,特别是微机械结构系统在实际中的应用越来越广泛。但是在微米或纳米量级下材料行为与宏观尺度下的理论预测完全不同,这就要求建立微尺度下的物理规律。
对偶应力理论进行了系统的阐述和总结,研究并推导了一般偶应力理论的平衡方程、应变位移关系和本构关系等。应变梯度理论是为解释材料在微米尺度下的尺寸效应现象而发展起来的一种新理论,偶应力理论是发展最早的一种应变梯度理论,现已在微观尺度的应力变形分析中得到广泛应用,但迄今主要依赖于数值计算方法,只有少数平面问题的解析解。
无网格法是近年来兴起的一种新的数值计算方法,在处理考虑偶应力的微机械结构振动问题时具有独特的优势。将偶应力理论与无网格方法相结合,发展了一种基于偶应力理论的使用于微结构振动分析的无网格方法,建立了无网格方法求解结构振动问题的本构模型和控制方程。
以基于偶应力理论的无网格方法为基础,研究微结构的振动问题,合理选取影响计算精度和收敛速度的几个无网格法重要参数,求出一般微结构的振动响应数值解,取得了理想的结果,为微机械结构设计和分析提供一定的理论依据。
最后,对基于偶应力理论的无网格方法在微机械振动领域的进一步应用做了展望。
The micromation of the mechanical devices and electronic devices is a trend of the development of industry.However, the behavior of material in the scale of micron meters is different from what the theory in macro meters has predicted.Therefore the theory for size effect was developed and had been used in the analysis of stress and strain of granular material, composite material, and the characteristic length of material was introduced in the constitutive relations of the theory.
The mechanics character of the couple stress theory is analyzed systemically. The equilibrium relations and the constitutive relations are derived for the general couple stress. Strain gradient deformation theory was developed in order to explain such size effect for materials in the scale of micron meters. The general couple stress theory is one of the strain gradient theories. There are many material constants in the coupled elastic body than the classic elastic body, it is difficult to obtain the analytic solutions, the numerical solution is necessary with the development of computer.
The mesh-free method based on the message of nodes has many advantages in analysis of micro-structure. A new mesh-free method for vibration analysis of micro-structure is developed, with couple stress theory combined with mesh-free method. The equilibrium relations and the constitutive relations are derived for the mesh-free method with couple stress theory.
Based on the mesh-free method with couple stress theory, the range of parameters in mesh-free method which have great impact on the result had been discussed. Solved the problems in vibration analysis of micro-structure, through the numerical example, the high accuracy, and high efficiency of the method were proved.
Lastly, the development of the mesh-free method in vibration analysis based on couple stress theory is forecasted.
对偶应力理论进行了系统的阐述和总结,研究并推导了一般偶应力理论的平衡方程、应变位移关系和本构关系等。应变梯度理论是为解释材料在微米尺度下的尺寸效应现象而发展起来的一种新理论,偶应力理论是发展最早的一种应变梯度理论,现已在微观尺度的应力变形分析中得到广泛应用,但迄今主要依赖于数值计算方法,只有少数平面问题的解析解。
无网格法是近年来兴起的一种新的数值计算方法,在处理考虑偶应力的微机械结构振动问题时具有独特的优势。将偶应力理论与无网格方法相结合,发展了一种基于偶应力理论的使用于微结构振动分析的无网格方法,建立了无网格方法求解结构振动问题的本构模型和控制方程。
以基于偶应力理论的无网格方法为基础,研究微结构的振动问题,合理选取影响计算精度和收敛速度的几个无网格法重要参数,求出一般微结构的振动响应数值解,取得了理想的结果,为微机械结构设计和分析提供一定的理论依据。
最后,对基于偶应力理论的无网格方法在微机械振动领域的进一步应用做了展望。
The micromation of the mechanical devices and electronic devices is a trend of the development of industry.However, the behavior of material in the scale of micron meters is different from what the theory in macro meters has predicted.Therefore the theory for size effect was developed and had been used in the analysis of stress and strain of granular material, composite material, and the characteristic length of material was introduced in the constitutive relations of the theory.
The mechanics character of the couple stress theory is analyzed systemically. The equilibrium relations and the constitutive relations are derived for the general couple stress. Strain gradient deformation theory was developed in order to explain such size effect for materials in the scale of micron meters. The general couple stress theory is one of the strain gradient theories. There are many material constants in the coupled elastic body than the classic elastic body, it is difficult to obtain the analytic solutions, the numerical solution is necessary with the development of computer.
The mesh-free method based on the message of nodes has many advantages in analysis of micro-structure. A new mesh-free method for vibration analysis of micro-structure is developed, with couple stress theory combined with mesh-free method. The equilibrium relations and the constitutive relations are derived for the mesh-free method with couple stress theory.
Based on the mesh-free method with couple stress theory, the range of parameters in mesh-free method which have great impact on the result had been discussed. Solved the problems in vibration analysis of micro-structure, through the numerical example, the high accuracy, and high efficiency of the method were proved.
Lastly, the development of the mesh-free method in vibration analysis based on couple stress theory is forecasted.
引文
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[3]康新,席占稳.基于Cosserat理论的微梁振动特性的尺度效应[J].机械强度,2007,(01)
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Materials, Cardiff, UK,2001.
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[17]Fleck N A, Huchinson J W.A phenomenological theory for strain gradient effects in plasticity [J]. J Mech Phys Solids,1993,41:1825-1857.
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[19]Shu J Y, Fleck N A. Strain gradient crystal plasticity:size-dependent deformation of bicrystals [J]. J Mech Phys Solids,1999,7:297-324.
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