Meshfree Method For The High-order Elastic-plastic Constitutive Model Based On The Couple Stress Theory

Because the high-order continuity theory includes the intrinsic characteristic length of materials,which can reflect the influence of material scale effect or micro-scale characteristics on macro-performance,and can be used to explain the phenomenon of scale effect,the high-order continuity theory is more and more used in various disciplines,but the corresponding numerical simulation methods are relatively few.Because the effect of strain gradient(the second derivative of displacement)needs to be considered,the numerical simulation of high-order elastoplastic problems is much more complicated than the traditional elastoplastic problems.The second derivative of displacement is not only an explicit high-order strain,but also included in yield criterion,which brings great trouble to numerical simulation.In this paper,the application of meshfree method in numerical simulation of high-order elastic-plastic problems is studied based on the high-order elastic-plastic constitutive model of couple stress theory.The main contents of this paper include:(1)On the basis of the couple stress theory and the high-order elastoplastic model based on the couple stress theory,the explicit expression of the couple stress elastoplastic increment matrix for the simplified plane problem and the concrete deduction process are given.In order to facilitate the establishment of a non-linear numerical calculation framework,the meshfree method is used for numerical discretization,and the generalized equivalent stress including the influence of higher-order stress and scale factor is defined and simplified,so that the meshfree method of high-order elastic-plastic model based on couple stress theory can be established by applying variational principle.(2)Moving least squares approximation is used to construct meshfree function.Moving least squares approximation function has the advantage of high order continuity.Displacement,strain and curvature are directly interpolated by node displacement value through shape function and its first and second derivatives.Thus,a high order elastoplastic meshfree numerical calculation framework and a nonlinear iterative algorithm based on couple stress theory are established.Fortran calculation program is compiled.(3)The two-dimensional cantilever beam and simply supported beam are numerically simulated by using the established meshfree method and the Fortran calculation program,which proves the validity of the established meshfree method.The influence of the size of the node influence area on the accuracy of the solution is analyzed,and the influence law of the scale factor on the structural deformation is studied.