The Element-Free/Mesh-Free Methods Based On The Moving Least-Squares Approximation And Their Application In The Numerical Simulations Of Higher-Order Continuum Structures

In recent years,the higher order continuum theory,such as strain gradient theory and couple stress theory,has been widely used in the analysis of micro or nanostructures,as well as the theoretical analysis of the macro structures considering the effect of micro-structure features.In these continuum theories,it is necessary to consider the influence of the higher order derivatives of displacements besides the traditional strain(the first derivative of the displacement).By introducing the angle(the first derivative of deflection)and curvature(the second derivative of deflection),two-dimensional beam can be simplified to a one-dimensional problem,the three-dimensional plate is simplified as a two-dimensional problem,because the angle and curvature,are coexisting basic variables,they can be regarded as a simple high order continuous structure.When considering the gradient effect of the beam or the plate,the beam or plate model based on the higher order continuum theory needs to consider the four or higher derivatives of the deflection.Numerical simulations of high order continuous structure may bring many new problems,for example,we need construct the C1 element in the finite element method while the angle or strain gradient as the basic unknown,and the computational cost is greatly increased.If we construct C2 continuous element or higher order continuous characteristics element,the computational cost will become larger.In the numerical method of other types(e.g.the boundary integral equation method),when the displacement and its higher derivative are coexisting basic variables,will also encounter the above problems,and they are collectively called as high order continuous structure in this paper.Compared with the traditional finite element method,element-free or mesh-free method has been developed in recent years and has some new advantages.The moving least squares(MLS)approximation has high order continuous features which brings great convenience to the numerical simulation of high order continuous structure as it can be used to construct high order continuous deformation field easily.This paper studied the element-free or mesh-free methods based on the moving least squares approximation and their application in the numerical simulations of higher-order continuum structures.The main research work is as follows:(1)An efficient boundary element-free method is presented for 2-D crack problems by combining a pair of boundary integral equations and the moving-least square approximation.The displacement boundary integral equation is collated on the on-crack boundary,and a new traction boundary integral equation is applied on the crack surface without the separate consideration of the upper and lower sides.In virtue of integration by parts,only singularity in order 1/r is involved in the integral kernels of new traction boundary integral equation,which brings convenience to the numerical implementation.Meanwhile,the integration by parts produces the new variables,the displacement density and displacement dislocation density,and they are the coexisting unknowns along with the displacement and displacement dislocation.With the high-order continuity of the moving-least square approximation,these new variables are directly approximated with the nodal displacement or displacement dislocation,and the final system of equations contains the unknowns of nodal displacements and displacement dislocations only.With the new boundary element-free method,the problems of straight crack in a square plate,a kinked crack a rectangular plate,two collinear cracks in an infinite plate and circular-arc crack in an unbounded domain are solved and the results show that the proposed method is highly efficient and flexible.(2)The kirchhoff plate is regarded as a simple high order continuous structure in this paper since the deflection,angle and curvature are basic variables.A mesh-free computational scheme for kirchhoff plate is established in which the basic variables are directly approximated with the nodal deflections in virtue of the higher-order continuum property of the shape function.The effects of nodal distribution and influence domain on the computational accuracy is studied.(3)A mesh-free method to the simulation of kirchhoff plate is presented in the scheme of the strain gradient theory in which an intrinsic material length is introduced to capture the size effect.The established thin plate theory introduces two order,three order and four order derivatives of the deflection.The moving least-square approximation is adopted to produce the shape function and its high-order derivatives,and some key factors of the numerical method are analyzed by several examples.The results show that when high order derivatives are considered,a larger influence domain is required to obtain good calculation accuracy.(4)A mesh-free method based on couple stress theory is established using the the moving least squares approximation(MLS).The cantilever beam with size effect and thin plate with a hole are analysied using the proposed method,and the variational rule on bending stiffness of the cantilever beam with the internal length of the micro-structure is obtained,and the strain distribution rule around structural cavities is analysied.(5)The problem of the cracking is still a challenging task since it involves some key issues such as simulation of the discontinuous displacement field,the treatment of singular region in crack tip and crack formation criterion,etc.A mesh-free method to the simulation of the cracking is presented that combines the cohesive zone model and crack segment model,in which cracking are regraded as the appearance and combination of series of cohesive segments.The results show that the proposed method is more efficient for modeling of cracking paths.