应变梯度弹性理论C~1自然单元法及其应用研究
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摘要
许多微观实验表明,当变形特征长度在微米或者亚微米量级时,材料表现出强烈的尺寸效应。经典理论的本构关系中不包含任何尺度参数,无法解释在微米和亚微米量级实验中发现的尺寸效应现象。为了解释微构件尺寸效应现象和解决工程实践中遇到的微尺寸设计问题,人们建立了应变梯度理论。应变梯度理论考虑了应变梯度高阶张量对应变能密度函数的影响,在本构关系中引入了材料的特征长度尺寸,能够描述和解释微构件力学性能的尺寸效应现象。应变梯度理论的控制方程和边界条件非常复杂,直接求取解析解非常困难,因此数值方法是解决这类边值问题的重要手段。应变梯度理论有限元法直接构建C1连续单元比较困难,为了降低连续性要求,通常采用C0连续混合单元法,该方法存在自由度多、计算工作量大等弊端。应变梯度理论无网格法的优点在于容易构建高阶连续形函数,然而,基于移动最小二乘法的无网格法,形函数的构造需要矩阵的求逆运算,计算效率较低,另外,形函数不具有插值特性,引入本质边界条件需要作特殊处理。针对目前应变梯度理论无网格法存在的不足,构建一种高效、高精度和易于处理本质边界条件的应变梯度理论无网格法,能够为模拟微构件力学性能的尺寸效应提供更为有效的数值计算手段。论文主要内容和研究成果包括：
将C1自然邻近插值函数和基于移动最小二乘法的形函数应用于曲面拟合场合,对它们在计算精度、计算效率和数值稳定性等方面的区别进行了定量分析和比较。
将C1自然单元法应用于偶应力弹性理论,建立了偶应力弹性理论C1自然单元法,由于形函数的插值性,本质边界条件的处理非常简单。为了考察构建方法的收敛速度和计算精度,分析了具有理论解的方块单剪问题和中心圆孔无限大板应力集中问题。对于方块单剪问题,随着离散结点的增多,数值解很快收敛于理论解；由于Voronoi结构能够自动调整结点分布不规则在空间上的差异,规则结点和不规则结点情况下计算结果差别不大,数值稳定性较好。对于小孔应力集中问题,计算结果表明偶应力对小孔应力分布的影响与无限大板的受力情况有关,纯剪情况下的影响要比单轴拉伸情况下的影响大,双轴拉伸时则没有影响。偶应力的作用范围局限于小孔附近,远离小孔,其影响可以忽略不计。算例的数值解与理论解吻合得较好,表明构建方法具有良好的计算精度。
将C1自然单元法应用于全应变梯度弹性理论,建立了全应变梯度弹性理论C1自然单元法,由于形函数的插值性,可以直接施加本质边界条件对于接触双材料界面边界层效应问题,随着结点的增加,数值解很快收敛于理论解,Voronoi结构能够自动调整结点分布不均匀在空间上的差异,均匀结点和非均匀结点情况下的计算结果差别也不大,数值稳定性较好。对于开孔无限大板承受双轴拉伸时,当考虑应变梯度效应时,高阶应力对应力分布产生了影响,降低了小孔的应力集中系数。算例的数值解与理论解吻合得较好,表明构建方法的计算精度较高。
将应变梯度弹性理论C1自然单元法应用于MEMS中,分析了一些结构和受力较为复杂并且不易求得理论解的实际工程问题。以微夹持器的夹持臂和中心开孔微试件为研究对象,研究了考虑应变梯度效应时微弹簧结构尺寸对夹持臂弯曲刚度、微梁厚度对驱动电压的影响规律以及开孔形状和尺寸对微试件应力集中系数的影响规律,探讨了它们对材料特征长度的尺寸依赖性,计算结果能够为微构件的设计和实验研究提供依据。
Many microscale experiments have shown that materials display strong size effects when the characteristic length scale is on the order of microns or submicrons. Traditional theories are unable to explain the size effects which have been observed in the micro-scales and submicro-scales experiments. One remedy against the deficiency of the traditional theories is to use the strain gradient theories whereby the strain energy density function depends on both the strain tensors and the strain gradient tensors. As the characteristic length scales of materials are introduced in the constitutive equations, strain gradient theories can predict the size effects. The governing equations and the boundary conditions of strain gradient theories are more complicated than those in traditional theories; therefore, numerical methods are usually effective ways to solve these boundary value problems (BVPs). There are some shortcomings in FEM for strain gradient theories, for example, it is difficult to construct the higher-order continuous elements, some C0 continuous mixed elements exist extra nodal degrees of fromdom (DOFs), and the computational effiency is low. It is easy to construct higher-order continuous shape functions in meshless methods for strain gradient theories. However, for meshless methods based on Moving Least-square Method (MLS), there are many inverse matrixes are included, so that, the construction of shape functions is complicated and the computational cost is high. Moreover, the shape functions lack the Kronecker delta properties, so that it is not easy to deal with the essential boundary conditions (EBCs). In order to model the size effets of microstructures effectively, it is necessay to construct the meshless method possessing high accuracy, high effiency and interpolating property.
In order to compare the discrepancies in computational accuracy, computational efficiency and interpolating property, C1 natural neighbor interpolant and shape function constructed by MLS are applied to surface fitting.
C1 natural element method for the solution of couple-stress elasticity is proposed. The shape functions have the interpolation to nodal function and nodal gradient values, so that EBCs can be imposed directly. In order to examine the convergence and computational accuracy, simple shear problem of the block and the infinite plate with a central circular hole subjected to the unaxial tension, biaxial tension and pure shear are analyzed. For the simple shear problem, with the increase of nodes, the numerical solutions converge quickly to the exact solutions. For the irregular and regular nodes, there is little change in the solutions because Voronoi digram can adjust discrepancy caused by the nodes automatically. For the stress concentration problem of the hole, the effects of couple stresses on the distribution of stresses are related with the loads. In fact, the effect of couple stresses in the case of pure shear is greater than that in the case of simple tension; couple stresses have no effect at all on the stressses in a field of isotropic tension. Couple stresses influence on small area around the hole, when far from the hole; the solution in the couple-stress elasticity is close to that in the traditional theory.
C1 natural element method for the solution of strain gradient elasticity is proposed. The shape functions have the interpolation to nodal function and nodal gradient values, so that it is easy to deal with EBCs. In order to examine the convergence and computational accuracy, boundary layer analysis of a bimaterial system and stress concentration due to a hole are analyzed. For the bimaterial system, with increase of nodes, the numerical solutions converge quickly to the exact solutions. For the non-uniform and uniform nodes, there is little change in the solutions because Voronoi digram can adjust discrepancy caused by the nodes automatically. For stress concentration due to a hole, the double stresses minish the stress concentration around the hole.
The validity and accuracy of the proposed methods are investigated through the numerical examples, and the numerical solutions are in good agreement with the analytical ones, which show that C1 natural element method can be used to analyze the couple-stress elasticity and strain gradient elasticity problems.
In the application of the proposed methods to MEMS, some pratical problems in engineering are analyzed. In these problems, typical components such as microgripper and microspeciem are taken as research objects, the impact of microspring's size on the bending stiffness, the impact of thickness of microbeam on voltage and the impact of the shape and size of hole on stress concentration factor are studied when considering the strain gradient effects. The dependent relations of these parameters on the characteristic length scales of materials are discussed. Numerical results can be used to provide some value evidence for design and experimental research.
将C1自然邻近插值函数和基于移动最小二乘法的形函数应用于曲面拟合场合,对它们在计算精度、计算效率和数值稳定性等方面的区别进行了定量分析和比较。
将C1自然单元法应用于偶应力弹性理论,建立了偶应力弹性理论C1自然单元法,由于形函数的插值性,本质边界条件的处理非常简单。为了考察构建方法的收敛速度和计算精度,分析了具有理论解的方块单剪问题和中心圆孔无限大板应力集中问题。对于方块单剪问题,随着离散结点的增多,数值解很快收敛于理论解；由于Voronoi结构能够自动调整结点分布不规则在空间上的差异,规则结点和不规则结点情况下计算结果差别不大,数值稳定性较好。对于小孔应力集中问题,计算结果表明偶应力对小孔应力分布的影响与无限大板的受力情况有关,纯剪情况下的影响要比单轴拉伸情况下的影响大,双轴拉伸时则没有影响。偶应力的作用范围局限于小孔附近,远离小孔,其影响可以忽略不计。算例的数值解与理论解吻合得较好,表明构建方法具有良好的计算精度。
将C1自然单元法应用于全应变梯度弹性理论,建立了全应变梯度弹性理论C1自然单元法,由于形函数的插值性,可以直接施加本质边界条件对于接触双材料界面边界层效应问题,随着结点的增加,数值解很快收敛于理论解,Voronoi结构能够自动调整结点分布不均匀在空间上的差异,均匀结点和非均匀结点情况下的计算结果差别也不大,数值稳定性较好。对于开孔无限大板承受双轴拉伸时,当考虑应变梯度效应时,高阶应力对应力分布产生了影响,降低了小孔的应力集中系数。算例的数值解与理论解吻合得较好,表明构建方法的计算精度较高。
将应变梯度弹性理论C1自然单元法应用于MEMS中,分析了一些结构和受力较为复杂并且不易求得理论解的实际工程问题。以微夹持器的夹持臂和中心开孔微试件为研究对象,研究了考虑应变梯度效应时微弹簧结构尺寸对夹持臂弯曲刚度、微梁厚度对驱动电压的影响规律以及开孔形状和尺寸对微试件应力集中系数的影响规律,探讨了它们对材料特征长度的尺寸依赖性,计算结果能够为微构件的设计和实验研究提供依据。
Many microscale experiments have shown that materials display strong size effects when the characteristic length scale is on the order of microns or submicrons. Traditional theories are unable to explain the size effects which have been observed in the micro-scales and submicro-scales experiments. One remedy against the deficiency of the traditional theories is to use the strain gradient theories whereby the strain energy density function depends on both the strain tensors and the strain gradient tensors. As the characteristic length scales of materials are introduced in the constitutive equations, strain gradient theories can predict the size effects. The governing equations and the boundary conditions of strain gradient theories are more complicated than those in traditional theories; therefore, numerical methods are usually effective ways to solve these boundary value problems (BVPs). There are some shortcomings in FEM for strain gradient theories, for example, it is difficult to construct the higher-order continuous elements, some C0 continuous mixed elements exist extra nodal degrees of fromdom (DOFs), and the computational effiency is low. It is easy to construct higher-order continuous shape functions in meshless methods for strain gradient theories. However, for meshless methods based on Moving Least-square Method (MLS), there are many inverse matrixes are included, so that, the construction of shape functions is complicated and the computational cost is high. Moreover, the shape functions lack the Kronecker delta properties, so that it is not easy to deal with the essential boundary conditions (EBCs). In order to model the size effets of microstructures effectively, it is necessay to construct the meshless method possessing high accuracy, high effiency and interpolating property.
In order to compare the discrepancies in computational accuracy, computational efficiency and interpolating property, C1 natural neighbor interpolant and shape function constructed by MLS are applied to surface fitting.
C1 natural element method for the solution of couple-stress elasticity is proposed. The shape functions have the interpolation to nodal function and nodal gradient values, so that EBCs can be imposed directly. In order to examine the convergence and computational accuracy, simple shear problem of the block and the infinite plate with a central circular hole subjected to the unaxial tension, biaxial tension and pure shear are analyzed. For the simple shear problem, with the increase of nodes, the numerical solutions converge quickly to the exact solutions. For the irregular and regular nodes, there is little change in the solutions because Voronoi digram can adjust discrepancy caused by the nodes automatically. For the stress concentration problem of the hole, the effects of couple stresses on the distribution of stresses are related with the loads. In fact, the effect of couple stresses in the case of pure shear is greater than that in the case of simple tension; couple stresses have no effect at all on the stressses in a field of isotropic tension. Couple stresses influence on small area around the hole, when far from the hole; the solution in the couple-stress elasticity is close to that in the traditional theory.
C1 natural element method for the solution of strain gradient elasticity is proposed. The shape functions have the interpolation to nodal function and nodal gradient values, so that it is easy to deal with EBCs. In order to examine the convergence and computational accuracy, boundary layer analysis of a bimaterial system and stress concentration due to a hole are analyzed. For the bimaterial system, with increase of nodes, the numerical solutions converge quickly to the exact solutions. For the non-uniform and uniform nodes, there is little change in the solutions because Voronoi digram can adjust discrepancy caused by the nodes automatically. For stress concentration due to a hole, the double stresses minish the stress concentration around the hole.
The validity and accuracy of the proposed methods are investigated through the numerical examples, and the numerical solutions are in good agreement with the analytical ones, which show that C1 natural element method can be used to analyze the couple-stress elasticity and strain gradient elasticity problems.
In the application of the proposed methods to MEMS, some pratical problems in engineering are analyzed. In these problems, typical components such as microgripper and microspeciem are taken as research objects, the impact of microspring's size on the bending stiffness, the impact of thickness of microbeam on voltage and the impact of the shape and size of hole on stress concentration factor are studied when considering the strain gradient effects. The dependent relations of these parameters on the characteristic length scales of materials are discussed. Numerical results can be used to provide some value evidence for design and experimental research.
引文
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