功能梯度材料平面问题的辛弹性力学解法
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摘要
功能梯度材料(FGM)是一种多相复合的非均匀材料,由于宏观特性沿空间坐标连续变化而具有优越的物理力学性能,在航空、航天、电子、机械、光学等多个领域获得广泛的应用。将梯度变化的概念引入压电和压电压磁材料产生了功能梯度压电材料(FGPM)和功能梯度压电压磁材料(FGMEEM),进一步推动了材料科学和智能结构的发展。
本文将辛弹性力学方法推广应用于功能梯度弹性、压电和压电压磁材料平面问题的力学分析。假设材料特性沿长度方向指数形式分布,通过引入新的应力(电位移和磁感应强度)分量,构造出新的状态向量,并建立相应的状态方程,进而利用分离变量和本征展开的方法对相关问题进行了探讨。
与均匀材料不同,FGM、FGPM和FGMEEM平面问题的算子矩阵并非哈密顿矩阵,本文提出了移位Hamilton矩阵的新概念,并利用其性质对本征值和本征解问题作了分析。通过求解发现,零本征值的本征解为圣维南问题的基本解；特殊本征值-α的本征解由于材料非均匀性的影响而发生明显的变化,其对应原问题的解去除刚体运动、常电势或常磁势后描述圣维南问题的解；般本征值对应原问题的解描述了边界效应的影响和随距离指数衰减的过程。
将辛方法用于求解双向功能梯度材料(2D-FGM)平面问题时,假设材料参数沿长度和厚度方向均按指数形式变化,算子矩阵表现出一些特殊的性质。对特殊本征值(零和-α)进行分析得到了圣维南问题的解,并且本征值-α的Jordan型本征向量的求解方法与均匀材料及单向FGM情形不同。通过求解发现,一般本征值对应原问题的解不再表现为简单对称和反对称变形状态,且材料非均匀性对其具有较大的影响作用。
按照传统的本征解展开式,当采用较多的项数时总会遇到数值计算不稳定的现象。文中对展开式中的一类本征解进行改写,可以在计算过程中避免大数的出现,从而有效改善了辛弹性力学方法中的数值计算稳定性。
本文工作表明,辛求解方法可应用于求解功能梯度材料结构问题,其进一步发展和完善将极大地丰富非均匀材料结构的分析手段。
As a special kind of inhomogeneous multiphase composites, functionally graded materials (FGMs) with material constants varying continuously with spatial positions exhibit excellent physical and mechanical properties, and hence they have been widely used in aeronautics and astronautics, electronics, machine, optics and so on. Functionally graded piezoelectric materials (FGPMs) and functionally graded magneto-electric-elastic materials (FGMEEMs) have also been developed by integrating the concept of FGM with piezoelectric and magneto-electro-elastic materials. These new types of functional materials promote a prospective development and application in material science and smart structures.
This paper extends the symplectic approach to investigate the plane problems of functionally graded elastic, piezoelectric and magneto-electric-elastic materials. The material properties are assumed to vary along the length direction in an identical exponential form. By introducing the new components of stress (electric displacement and magnetic induction), the modified state vector is constructed and the state equation is derived. Within the symplectic framework, the origin problem can be converted into analyzing the eigenvalues and eigensolutions through the method of separation of variables along with the eigenfunction expansion technique.
In contrast to the homogenous materials, the operator matrices for FGM, FGPM and FGMEEM are not in a conventional Hamiltonian form. In this paper, a new concept, i.e. the Shift-Hamiltonian operator, is coined since its eigenvalues are symmetric with respect to-α/2, instead of zero as for the conventional Hamiltonian matrix. The zero eigensolutions bear definite physical interpretations; while the origin solutions corresponding to the particular eigenvalue -a exhibit some unique characteristics, which can not describe the Saint-Venant solutions directly because of the influence of material inhomogeneity. The origin solutions corresponding to other general eigenvalues (i.e. those excluding the particular eigenvalues zero and-α) represent the local effect and the decay exponentially with the distance.
The symplectic method is also applied to analyze plane problems of two-dimensional functionally graded materials (2D-FGMs), whose elastic modulus varies exponentially both along the axial and transverse coordinates while the Poisson's ratio remains constant. The operator matrix exhibits somehow different characteristics when compared with the conventional Hamiltonian matrix as well as the Shift-Hamiltonian matrix. From the physical point of view, it is known that the Saint-Venant solutions still correspond to the particular eigenvalues zero and-α. The first-order eigenvector of Jordan normal form for the special eigenvalue-αis solved in a different way than that for the homogenous materials and the 1D axial FGM. The material inhomogeneity has an important effect on the general eigenvalues.
According to the traditional formulation of the symplectic expansion series, enormous difference in magnitudes exists between two sets of general eigensolutions when a large number of eigensolutions are adopted to obtain accurate results. The large numbers can be avoided by rewriting the expansion formula so as to achieve the stability of numerical calculation. The numerical examples show that the treatment is simple and effective for both homogenous and functionally graded materials.
Through the present study, it is seen that the symplectic method can be an alternative to solve problems of functionally graded materials. Its further development will greatly enrich the current analysis methodology for heterogeneous materials and structures.
本文将辛弹性力学方法推广应用于功能梯度弹性、压电和压电压磁材料平面问题的力学分析。假设材料特性沿长度方向指数形式分布,通过引入新的应力(电位移和磁感应强度)分量,构造出新的状态向量,并建立相应的状态方程,进而利用分离变量和本征展开的方法对相关问题进行了探讨。
与均匀材料不同,FGM、FGPM和FGMEEM平面问题的算子矩阵并非哈密顿矩阵,本文提出了移位Hamilton矩阵的新概念,并利用其性质对本征值和本征解问题作了分析。通过求解发现,零本征值的本征解为圣维南问题的基本解；特殊本征值-α的本征解由于材料非均匀性的影响而发生明显的变化,其对应原问题的解去除刚体运动、常电势或常磁势后描述圣维南问题的解；般本征值对应原问题的解描述了边界效应的影响和随距离指数衰减的过程。
将辛方法用于求解双向功能梯度材料(2D-FGM)平面问题时,假设材料参数沿长度和厚度方向均按指数形式变化,算子矩阵表现出一些特殊的性质。对特殊本征值(零和-α)进行分析得到了圣维南问题的解,并且本征值-α的Jordan型本征向量的求解方法与均匀材料及单向FGM情形不同。通过求解发现,一般本征值对应原问题的解不再表现为简单对称和反对称变形状态,且材料非均匀性对其具有较大的影响作用。
按照传统的本征解展开式,当采用较多的项数时总会遇到数值计算不稳定的现象。文中对展开式中的一类本征解进行改写,可以在计算过程中避免大数的出现,从而有效改善了辛弹性力学方法中的数值计算稳定性。
本文工作表明,辛求解方法可应用于求解功能梯度材料结构问题,其进一步发展和完善将极大地丰富非均匀材料结构的分析手段。
As a special kind of inhomogeneous multiphase composites, functionally graded materials (FGMs) with material constants varying continuously with spatial positions exhibit excellent physical and mechanical properties, and hence they have been widely used in aeronautics and astronautics, electronics, machine, optics and so on. Functionally graded piezoelectric materials (FGPMs) and functionally graded magneto-electric-elastic materials (FGMEEMs) have also been developed by integrating the concept of FGM with piezoelectric and magneto-electro-elastic materials. These new types of functional materials promote a prospective development and application in material science and smart structures.
This paper extends the symplectic approach to investigate the plane problems of functionally graded elastic, piezoelectric and magneto-electric-elastic materials. The material properties are assumed to vary along the length direction in an identical exponential form. By introducing the new components of stress (electric displacement and magnetic induction), the modified state vector is constructed and the state equation is derived. Within the symplectic framework, the origin problem can be converted into analyzing the eigenvalues and eigensolutions through the method of separation of variables along with the eigenfunction expansion technique.
In contrast to the homogenous materials, the operator matrices for FGM, FGPM and FGMEEM are not in a conventional Hamiltonian form. In this paper, a new concept, i.e. the Shift-Hamiltonian operator, is coined since its eigenvalues are symmetric with respect to-α/2, instead of zero as for the conventional Hamiltonian matrix. The zero eigensolutions bear definite physical interpretations; while the origin solutions corresponding to the particular eigenvalue -a exhibit some unique characteristics, which can not describe the Saint-Venant solutions directly because of the influence of material inhomogeneity. The origin solutions corresponding to other general eigenvalues (i.e. those excluding the particular eigenvalues zero and-α) represent the local effect and the decay exponentially with the distance.
The symplectic method is also applied to analyze plane problems of two-dimensional functionally graded materials (2D-FGMs), whose elastic modulus varies exponentially both along the axial and transverse coordinates while the Poisson's ratio remains constant. The operator matrix exhibits somehow different characteristics when compared with the conventional Hamiltonian matrix as well as the Shift-Hamiltonian matrix. From the physical point of view, it is known that the Saint-Venant solutions still correspond to the particular eigenvalues zero and-α. The first-order eigenvector of Jordan normal form for the special eigenvalue-αis solved in a different way than that for the homogenous materials and the 1D axial FGM. The material inhomogeneity has an important effect on the general eigenvalues.
According to the traditional formulation of the symplectic expansion series, enormous difference in magnitudes exists between two sets of general eigensolutions when a large number of eigensolutions are adopted to obtain accurate results. The large numbers can be avoided by rewriting the expansion formula so as to achieve the stability of numerical calculation. The numerical examples show that the treatment is simple and effective for both homogenous and functionally graded materials.
Through the present study, it is seen that the symplectic method can be an alternative to solve problems of functionally graded materials. Its further development will greatly enrich the current analysis methodology for heterogeneous materials and structures.
引文
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