基于偶应力理论的局部彼得洛夫—伽辽金无网格（MLPG）法
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摘要
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理论。偶应力理论是应变梯度理论的一种。局部彼得洛夫-伽辽金无网格法在求解过程中不需要背景网格,是一种“真正”的无网格法。本文基于对偶应力理论和局部彼得洛夫-伽辽金无网格(MLPG)法的深入分析研究,将偶应力理论与MLPG有机的结合,发展了一种新型的无网格法,并在此基础上编写了MATLAB程序,通过实例验证了此方法的可行性及有效性。
本文主要研究内容和结论有:
对一般偶应力理论的力学行为进行了系统的描述。引入了独立的微观转动矢量,与位移矢量互相独立,这是一般偶应力理论与传统偶应力理论最重要的区别。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。
系统地研究分析了MLPG法,将控制方程的子域积分“弱”形式引入到该方法中,使得对试探函数的连续性要求降低了,对实际问题能给出更好的近似解。这一方法中试探函数与测试函数可以取自不同的函数空间,子域构造形式多样,具有灵活多样的实施方案。MLPG法不需要额外的背景网格,刚度矩阵的形成不需要装配过程。
将偶应力理论的本构关系代入到MLPG法的离散方程中,发展了一种基于偶应力理论的新型的MLPG法。
利用本文推导出的方法对带中心小孔的无限平板在单轴拉伸及纯剪状态下的应力集中情况进行数值计算与分析。计算结果表明,偶应力的存在对圆孔周围应力集中起了缓解作用,当小孔半径与材料内禀长度相当时,应力集中因子小于经典弹性理论中的3,应力集中因子随α/ι的增大而增大,最终趋近于经典弹性理论解;并且应力集中因子还与泊松比有关,随泊松比的减小而减小。此外,本文还用该方法分析边界层问题。探讨了偶应力对层合板变形的影响,在边界层区域内,存在一定的转动梯度效应,考虑偶应力时,剪应变的分布是连续的;分析了不同布点方式对计算结果的影响,选择合适的布点方式是提高MLPG法计算精度的有效途径。本文推导出的基于偶应力理论的MLPG法为求解实际工程中具有尺寸效应的复杂结构的数值计算提供了一种强有力的方法。
最后,对基于一般偶应力理论的无网格法的应用及发展做了展望。
In order to explain the size effect for materials in the scale of micron meters,the strain gradient theory was developed.The general couple stress theory is one of the strain gradient theories.The meshless local Petrov-Galerkin(MLPG)method is a truly meshless method,as it does not need any "finite element mesh".Based on the analysis and research of the gmeral couple stress theory and MLPG method,the couple stress theory is combined with MLPG in this paper,and developed a new meshless method. MATLAB programs are compiled on this method,together with some classical experiments,proved its accuracy,feasibility and efficiency.
The main research content and conclusion are stated as followed:
The mechanics character of the general couple stress theory is analyzed systemically.The micro-rotation is introduced in this theory,which is treated as an independent kinematic quantity with no direct dependence upon the displacement vector. This is the most prominent difference between the general couple stress theory and the classical theory.The equilibrium relations and the constitutive relations are derived for the general couple stress.
The current study,application and development trends of meshless methods are summarized and analyzed in this paper.The approaximation,discretization,integration scheme and treatment of essential boundary conditions used in meshless methods are overviewed.MLPG method,using a local weak form of the equilibrium equations and shape functions from the moving least squares(MLS)approaximation,attracted much attention.As the trial and test functions can be chosen from different functional spaces and the construction types of the sub-domain is flexible,the MLPG method is a generalized method to construct different meshless methods as the trial and test functions and the integration schemes are selected appropriately.MLPG method is a truly meshless method and the assembly process of global stiffness matrix is avoided.A new MLPG method is derived,with the couple stress theory combined with MLPG method.
In order to deal with stress concentration problems,the problems of the circular hole in a field of uniform tension and compressions in two perpendicular directions are studied.The couple stress has great effect on the stress concentration around the circle. When couple stress must be considered,the stress concentration factor increases withα/l(hole radius/characteristic length)increases,and finally it approaches classical solution 3.The stress concentration factor also depends on Poisson's ration,decreasing with variations of Poisson's ration.
In addition to this,the boundary layer problem is studied.A bimaterial system composed of two perfectly bonded half planes of elastic strain gradient solids,subjected to a remote shear stress.For this bimaterial system,the conventional elasticity couple stress theory dictates that the shear stress is uniform;but the shear strain jumps in magnitude at the interface.By including the strain gradient effects,a continuously distributed shear strain can be obtained.The numerical solution converges quickly to the exact solution,with an increasing refinement of nodal distances.It is interesting to note that the strain caculated at the interface is accurate even for a coarse nodal pattern.
All of these show that the new MLPG method possesses several advantages,such as high accuracy,high stability,and high efficiency.
Lastly,the development of the MLPG method based on general couple stress theory is forecasted.
本文主要研究内容和结论有:
对一般偶应力理论的力学行为进行了系统的描述。引入了独立的微观转动矢量,与位移矢量互相独立,这是一般偶应力理论与传统偶应力理论最重要的区别。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。
系统地研究分析了MLPG法,将控制方程的子域积分“弱”形式引入到该方法中,使得对试探函数的连续性要求降低了,对实际问题能给出更好的近似解。这一方法中试探函数与测试函数可以取自不同的函数空间,子域构造形式多样,具有灵活多样的实施方案。MLPG法不需要额外的背景网格,刚度矩阵的形成不需要装配过程。
将偶应力理论的本构关系代入到MLPG法的离散方程中,发展了一种基于偶应力理论的新型的MLPG法。
利用本文推导出的方法对带中心小孔的无限平板在单轴拉伸及纯剪状态下的应力集中情况进行数值计算与分析。计算结果表明,偶应力的存在对圆孔周围应力集中起了缓解作用,当小孔半径与材料内禀长度相当时,应力集中因子小于经典弹性理论中的3,应力集中因子随α/ι的增大而增大,最终趋近于经典弹性理论解;并且应力集中因子还与泊松比有关,随泊松比的减小而减小。此外,本文还用该方法分析边界层问题。探讨了偶应力对层合板变形的影响,在边界层区域内,存在一定的转动梯度效应,考虑偶应力时,剪应变的分布是连续的;分析了不同布点方式对计算结果的影响,选择合适的布点方式是提高MLPG法计算精度的有效途径。本文推导出的基于偶应力理论的MLPG法为求解实际工程中具有尺寸效应的复杂结构的数值计算提供了一种强有力的方法。
最后,对基于一般偶应力理论的无网格法的应用及发展做了展望。
In order to explain the size effect for materials in the scale of micron meters,the strain gradient theory was developed.The general couple stress theory is one of the strain gradient theories.The meshless local Petrov-Galerkin(MLPG)method is a truly meshless method,as it does not need any "finite element mesh".Based on the analysis and research of the gmeral couple stress theory and MLPG method,the couple stress theory is combined with MLPG in this paper,and developed a new meshless method. MATLAB programs are compiled on this method,together with some classical experiments,proved its accuracy,feasibility and efficiency.
The main research content and conclusion are stated as followed:
The mechanics character of the general couple stress theory is analyzed systemically.The micro-rotation is introduced in this theory,which is treated as an independent kinematic quantity with no direct dependence upon the displacement vector. This is the most prominent difference between the general couple stress theory and the classical theory.The equilibrium relations and the constitutive relations are derived for the general couple stress.
The current study,application and development trends of meshless methods are summarized and analyzed in this paper.The approaximation,discretization,integration scheme and treatment of essential boundary conditions used in meshless methods are overviewed.MLPG method,using a local weak form of the equilibrium equations and shape functions from the moving least squares(MLS)approaximation,attracted much attention.As the trial and test functions can be chosen from different functional spaces and the construction types of the sub-domain is flexible,the MLPG method is a generalized method to construct different meshless methods as the trial and test functions and the integration schemes are selected appropriately.MLPG method is a truly meshless method and the assembly process of global stiffness matrix is avoided.A new MLPG method is derived,with the couple stress theory combined with MLPG method.
In order to deal with stress concentration problems,the problems of the circular hole in a field of uniform tension and compressions in two perpendicular directions are studied.The couple stress has great effect on the stress concentration around the circle. When couple stress must be considered,the stress concentration factor increases withα/l(hole radius/characteristic length)increases,and finally it approaches classical solution 3.The stress concentration factor also depends on Poisson's ration,decreasing with variations of Poisson's ration.
In addition to this,the boundary layer problem is studied.A bimaterial system composed of two perfectly bonded half planes of elastic strain gradient solids,subjected to a remote shear stress.For this bimaterial system,the conventional elasticity couple stress theory dictates that the shear stress is uniform;but the shear strain jumps in magnitude at the interface.By including the strain gradient effects,a continuously distributed shear strain can be obtained.The numerical solution converges quickly to the exact solution,with an increasing refinement of nodal distances.It is interesting to note that the strain caculated at the interface is accurate even for a coarse nodal pattern.
All of these show that the new MLPG method possesses several advantages,such as high accuracy,high stability,and high efficiency.
Lastly,the development of the MLPG method based on general couple stress theory is forecasted.
引文
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