The Meshless Local Petrov-Galerkin (MLPG) Method With The Couple Stress Theory

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.