偶应力/应变梯度理论的精化不协调元分析
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摘要
随着科学技术的发展,各种微／纳米器件的研究和应用日趋广泛。目前已有大量的力学实验表明：在微／纳观尺度下,材料的力学行为呈现出强烈的尺度效应。尺度效应的存在给微／纳米器件的结构设计提出了一系列新的挑战。经典连续介质力学的本构方程不包含任何与尺度相关的材料参数,所以不能预测尺度效应。梯度理论将具有长度量纲的材料长度参数引入本构模型,可以解释尺度效应,现已在微／纳观尺度下的金属材料、颗粒材料和复合材料等的力学行为研究中得到广泛应用。
本文主要研究的是梯度理论中具有代表性的两种理论：偶应力和应变梯度理论。偶应力／应变梯度理论较传统连续理论更为复杂,迄今只有少数问题获得了解析解,有限元方法就成为重要的分析手段。可靠的有限元法不仅是工程应用的需要,也是材料长度参数识别的需要,因此对有限元计算精度有很高的要求。偶应力／应变梯度理论同时包含位移的一阶和二阶导数,位移插值函数需满足C1连续条件。C1协调单元的节点参数含有位移的高阶导数,构造和应用都较为困难,并且数量十分有限。目前广泛采用的是C0单元,位移及位移梯度独立插值,它们之间的约束关系通过Lagrange乘子法或罚函数法满足。但是,Lagrange乘子法会增加计算量；而罚函数法的数值结果会受罚因子大小的影响。偶应力／应变梯度理论有限元在单元构造和检验单元收敛性方面都不够成熟。
相对于协调单元,不协调单元放松了单元间的连续条件,可以构造形式更为灵活的单元函数,从而更容易建立高精度的单元。目前已经建立的偶应力／应变梯度单元都只分别考虑满足C0连续或C1连续。本文基于精化不协调元方法,建立了两种分别用于平面问题和轴对称问题求解且同时满足C0连续(或弱连续)和c1弱连续的偶应力／应变梯度精化不协调单元。首先,本文建立了24-DOF的平面四边形偶应力／应变梯度精化不协调元(CQ12+RDKQ)。提出了一个放松单元间连续条件的扩展的Hu-Washizu变分原理,并在此基础上首次建立了一个满足C0弱连续且有二次完备性的12-DOF四边形不协调元CQ12,用于计算应变；应变梯度由满足C1弱连续的12-DOF薄板单元RDKQ计算。通过将板单元节点参数转换为平面单元节点参数,二者组合建立24-DOF平面偶应力／应变梯度精化不协调元(CQ12+RDKQ)。其次,本文建立了18-DOF的轴对称三角形偶应力／应变梯度精化不协调元(BCIZ+ART9)。目前已建立的轴对称偶应力／应变梯度单元较少,轴对称C1连续单元尚未建立。本文首次建立了轴对称不协调元的弱C1连续条件,进一步,建立了轴对称单元(BCIZ+ART9),其中BCIZ满足C0连续且具有二次完备性,用于计算位移的一阶导数,ART9满足本文建立的C1弱连续条件,用于计算位移的二阶导数。
有限元法分片检验是检验单元收敛性的实用准则。偶应力／应变梯度理论的控制方程属于非齐次微分方程,传统的分片检验函数不再适用。本文基于C0-1分片检验和增强型分片检验的思想建立了轴对称偶应力单元的分片检验函数,并证明了对于传统轴对称单元,不存在常剪力分片检验函数。本文建立的精化不协调元(CQ12+RDKQ)和(BCIZ+ART9)都能通过分片检验,收敛性得到保证,且有较高的效率和精度。
最后,应用本文建立的单元,通过钢筋拉拔弹性阶段和超薄悬臂梁受压弯曲问题的数值模拟,初步探讨了两种高阶导数项符号相反的应变梯度理论在描述材料尺度效应方面的区别。
There are extensive researches and applications of micro/nano-devices, along with the development of science and technology. At present, numerous experiments have demonstrated strong size effects on mechanical properties of materials at the micro/nano-scale. The micro/nano device design faces a series of new challenges posed by size effects. The classical continuum mechanics is not applicable for revealing the size effects of materials due to the lack of any material parameter corresponding to internal length scale in the constitutive models. Gradient theories can successfully explain the size effects by introducing the material length parameters with the length dimensions into the constitutive models, and have been widely used in the mechanical analysis of metal, granular and composite materials at the micro/nano-scale.
This paper is focused on the couple stress theory and the strain gradient theory, which are two kinds of typical and widely used gradient theories. Compared with the conventional continuum mechanics, the couple stress/strain gradient theory is substantially more complicated, and heretofore only a few analytical solutions are available. Finite element method provides an important approach. Reliable finite element method is needed not only for the purpose of engineering applications, but also for the identification of the material length parameters where higher order accuracy is required. In the finite element analysis, the displacement interpolation function should satisfy the requirement of C1 continuity as first and second derivatives of the displacement are involved in the couple stress/strain gradient theory. C1 conforming elements contain the nodal parameters with high order derivatives, and are complicated to construct and implement. Further, there are few C1 conforming elements available. Currently, the most widely used couple stress/strain gradient elements are C0 elements, in which displacements and displacement gradients are interpolated independently and their kinematic constraints are enforced via the penalty or Lagrange multiplier method. However, it is difficult to identify the penalty function, and the Lagrange multiplier method may increase the computation cost. The methods of finite element construction and convergence test for the couple stress/strain gradient theory have not been fully developed.
Compared with the conforming element methods, it is easier for the nonconforming element methods to establish high-performance elements as they relax the continuity condition more loosely and offer more flexible interpolation algorithms. The existing couple stress/strain gradient elements are constructed based on the consideration of either C0 or C1 continuity. In this paper, the refined nonconforming finite element methods are used to establish the plane and axisymmetric couple stress/strain gradient elements which satisfy C0 continuity (or weak C0 continuity) and C1 continuity simultaneously. Firstly, a 24-DOF (degrees of freedom) quadrilateral refined nonconforming element (CQ12+RDKQ) for the couple stress/strain gradient theory is developed. An extended Hu-Washizu variational principle which relaxes the continuity condition is proposed. Based on this variational principle, a 12-DOF quadrilateral nonconforming element CQ12 which satisfies weak C0 continuity and quadratic completeness is developed to calculate strains. The strain gradients are computed by the 12-DOF thin plate element RDKQ, which satisfies weak C1 continuity. By combining RDKQ and CQ12, and replacing the parameters of plate element by those of plane element, the 24-DOF element (CQ12+RDKQ) is established. Secondly, an 18-DOF axisymmetric triangular refined nonconforming element (BCIZ+ART9) for the couple stress/strain gradient theory is derived. Up to now, only a few axisymmetric couple stress/strain gradient elements have been developed. The axisymmetric C1 element does not exist. In this paper, a weak C1 continuity condition of axisymmetric nonconforming element method is proposed, and furthermore, the axisymmetric element (BCIZ+ART9) is developed. The displacement function of BCIZ, which satisfies C0 continuity and quadratic completeness, is used to calculate first derivatives of displacement. And the displacement function of ART9, which satisfies the proposed C1 weak continuity condition, is used to calculate second derivatives of displacement.
In finite element analysis, the patch test has been used as a criterion for assessing the convergence of finite elements. The equilibrium differential equations of couple stress/strain gradient theory are inhomogeneous, and the conventional patch test functions are not appropriate for such kind of problems. In this paper, based on the C0-1 patch test and the enhanced patch test, the patch test function for axisymmetric couple stress element is established, and furthermore, it is proved that the constant shear stress patch test function does not exist for conventional axisymmetric elements. The proposed elements (CQ12+RDKQ) and (BCIZ+ART9) can both pass the patch test (ensure convergence) and possess high-performance.
Finally, utilizing the proposed elements, the elastic process of the reinforcement pull-out and the deformation of a cantilever beam are simulated based on two kinds of strain gradient theories in which the signs of higher-order differential terms are opposite. The numerical results show the difference between the two theories in the aspect of describing size effects of materials.
本文主要研究的是梯度理论中具有代表性的两种理论：偶应力和应变梯度理论。偶应力／应变梯度理论较传统连续理论更为复杂,迄今只有少数问题获得了解析解,有限元方法就成为重要的分析手段。可靠的有限元法不仅是工程应用的需要,也是材料长度参数识别的需要,因此对有限元计算精度有很高的要求。偶应力／应变梯度理论同时包含位移的一阶和二阶导数,位移插值函数需满足C1连续条件。C1协调单元的节点参数含有位移的高阶导数,构造和应用都较为困难,并且数量十分有限。目前广泛采用的是C0单元,位移及位移梯度独立插值,它们之间的约束关系通过Lagrange乘子法或罚函数法满足。但是,Lagrange乘子法会增加计算量；而罚函数法的数值结果会受罚因子大小的影响。偶应力／应变梯度理论有限元在单元构造和检验单元收敛性方面都不够成熟。
相对于协调单元,不协调单元放松了单元间的连续条件,可以构造形式更为灵活的单元函数,从而更容易建立高精度的单元。目前已经建立的偶应力／应变梯度单元都只分别考虑满足C0连续或C1连续。本文基于精化不协调元方法,建立了两种分别用于平面问题和轴对称问题求解且同时满足C0连续(或弱连续)和c1弱连续的偶应力／应变梯度精化不协调单元。首先,本文建立了24-DOF的平面四边形偶应力／应变梯度精化不协调元(CQ12+RDKQ)。提出了一个放松单元间连续条件的扩展的Hu-Washizu变分原理,并在此基础上首次建立了一个满足C0弱连续且有二次完备性的12-DOF四边形不协调元CQ12,用于计算应变；应变梯度由满足C1弱连续的12-DOF薄板单元RDKQ计算。通过将板单元节点参数转换为平面单元节点参数,二者组合建立24-DOF平面偶应力／应变梯度精化不协调元(CQ12+RDKQ)。其次,本文建立了18-DOF的轴对称三角形偶应力／应变梯度精化不协调元(BCIZ+ART9)。目前已建立的轴对称偶应力／应变梯度单元较少,轴对称C1连续单元尚未建立。本文首次建立了轴对称不协调元的弱C1连续条件,进一步,建立了轴对称单元(BCIZ+ART9),其中BCIZ满足C0连续且具有二次完备性,用于计算位移的一阶导数,ART9满足本文建立的C1弱连续条件,用于计算位移的二阶导数。
有限元法分片检验是检验单元收敛性的实用准则。偶应力／应变梯度理论的控制方程属于非齐次微分方程,传统的分片检验函数不再适用。本文基于C0-1分片检验和增强型分片检验的思想建立了轴对称偶应力单元的分片检验函数,并证明了对于传统轴对称单元,不存在常剪力分片检验函数。本文建立的精化不协调元(CQ12+RDKQ)和(BCIZ+ART9)都能通过分片检验,收敛性得到保证,且有较高的效率和精度。
最后,应用本文建立的单元,通过钢筋拉拔弹性阶段和超薄悬臂梁受压弯曲问题的数值模拟,初步探讨了两种高阶导数项符号相反的应变梯度理论在描述材料尺度效应方面的区别。
There are extensive researches and applications of micro/nano-devices, along with the development of science and technology. At present, numerous experiments have demonstrated strong size effects on mechanical properties of materials at the micro/nano-scale. The micro/nano device design faces a series of new challenges posed by size effects. The classical continuum mechanics is not applicable for revealing the size effects of materials due to the lack of any material parameter corresponding to internal length scale in the constitutive models. Gradient theories can successfully explain the size effects by introducing the material length parameters with the length dimensions into the constitutive models, and have been widely used in the mechanical analysis of metal, granular and composite materials at the micro/nano-scale.
This paper is focused on the couple stress theory and the strain gradient theory, which are two kinds of typical and widely used gradient theories. Compared with the conventional continuum mechanics, the couple stress/strain gradient theory is substantially more complicated, and heretofore only a few analytical solutions are available. Finite element method provides an important approach. Reliable finite element method is needed not only for the purpose of engineering applications, but also for the identification of the material length parameters where higher order accuracy is required. In the finite element analysis, the displacement interpolation function should satisfy the requirement of C1 continuity as first and second derivatives of the displacement are involved in the couple stress/strain gradient theory. C1 conforming elements contain the nodal parameters with high order derivatives, and are complicated to construct and implement. Further, there are few C1 conforming elements available. Currently, the most widely used couple stress/strain gradient elements are C0 elements, in which displacements and displacement gradients are interpolated independently and their kinematic constraints are enforced via the penalty or Lagrange multiplier method. However, it is difficult to identify the penalty function, and the Lagrange multiplier method may increase the computation cost. The methods of finite element construction and convergence test for the couple stress/strain gradient theory have not been fully developed.
Compared with the conforming element methods, it is easier for the nonconforming element methods to establish high-performance elements as they relax the continuity condition more loosely and offer more flexible interpolation algorithms. The existing couple stress/strain gradient elements are constructed based on the consideration of either C0 or C1 continuity. In this paper, the refined nonconforming finite element methods are used to establish the plane and axisymmetric couple stress/strain gradient elements which satisfy C0 continuity (or weak C0 continuity) and C1 continuity simultaneously. Firstly, a 24-DOF (degrees of freedom) quadrilateral refined nonconforming element (CQ12+RDKQ) for the couple stress/strain gradient theory is developed. An extended Hu-Washizu variational principle which relaxes the continuity condition is proposed. Based on this variational principle, a 12-DOF quadrilateral nonconforming element CQ12 which satisfies weak C0 continuity and quadratic completeness is developed to calculate strains. The strain gradients are computed by the 12-DOF thin plate element RDKQ, which satisfies weak C1 continuity. By combining RDKQ and CQ12, and replacing the parameters of plate element by those of plane element, the 24-DOF element (CQ12+RDKQ) is established. Secondly, an 18-DOF axisymmetric triangular refined nonconforming element (BCIZ+ART9) for the couple stress/strain gradient theory is derived. Up to now, only a few axisymmetric couple stress/strain gradient elements have been developed. The axisymmetric C1 element does not exist. In this paper, a weak C1 continuity condition of axisymmetric nonconforming element method is proposed, and furthermore, the axisymmetric element (BCIZ+ART9) is developed. The displacement function of BCIZ, which satisfies C0 continuity and quadratic completeness, is used to calculate first derivatives of displacement. And the displacement function of ART9, which satisfies the proposed C1 weak continuity condition, is used to calculate second derivatives of displacement.
In finite element analysis, the patch test has been used as a criterion for assessing the convergence of finite elements. The equilibrium differential equations of couple stress/strain gradient theory are inhomogeneous, and the conventional patch test functions are not appropriate for such kind of problems. In this paper, based on the C0-1 patch test and the enhanced patch test, the patch test function for axisymmetric couple stress element is established, and furthermore, it is proved that the constant shear stress patch test function does not exist for conventional axisymmetric elements. The proposed elements (CQ12+RDKQ) and (BCIZ+ART9) can both pass the patch test (ensure convergence) and possess high-performance.
Finally, utilizing the proposed elements, the elastic process of the reinforcement pull-out and the deformation of a cantilever beam are simulated based on two kinds of strain gradient theories in which the signs of higher-order differential terms are opposite. The numerical results show the difference between the two theories in the aspect of describing size effects of materials.
引文
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