Finite Element Method For Plane Elastic Cosserat Bodies And Its Implementation

Recent years,some experiments have proved that there is strong size effect on the micro constructs when they come into the level of micron,that is to say,the mechanical property is significantly changed as the size of materials changed,which is unexplained in the macro mechanics.In classical continuum theory,the materials are continuous and uniform,such ideal treatment is suitable for both macro and micro structure,which does not consider the internal microscopic structure of materials,therefore the classical continuum theory can not explain the size effect of materials.Due to the introduction of strain gradient and internal characteristic length parameter for dimensional equilibrium,strain gradient theory can solve the problem of size effect.As one of the strain gradient theories,Cosserat elastic theory includes the rotation freedom on every material point,the gradient of rotation freedom and its work conjugate quantity couple stress are also introduced accordingly.There are only a few questions on Cosserat theory that have analytical solutions,so it becomes the effective way to solve the questions on Cosserat theory that get the numerical solution.Here the finite element method is applied to solve the boundary value problem on elastic Cosserat bodies,the basis equations are introduced first,then the variational equation is derived,and finite element formulations are also introduced.The finite element method based Cosserat theory is realized by the famous FEM software ABAQUS user subroutine UEL and UVARM.To validate the finite element method,the stress concentration problem of a circle hole are numerically simulated,the numerical results indicate that the stress concentration factors based on Cosserat theory FEM are much closer to which based on Cosserat theory.It can be verified that FEM based on ABAQUS secondary development is effective.A 4-nodes general isoparametric finite element is developed to deal with the problems considering the plane elastic Cosserat bodies in this paper.The additional coupling terms are introduced in the shape functions,which can be constructed by variational principle of displacement,and the element equations are given.The cantilever beam under shearing force is analyzed to check the property of the new element;the results indicate that the new element has high accuracy.The calculation accuracy of the elements are improved,however the degrees of freedom remain the same.The new element just adds the additional coupling terms based on the shape function of the conditional isoparametric finite elements;as a result the new element is convenient for implementation.Based on area coordinate method,a new element is constructed.Numerical examples show that the new element formulated by these new natural coordinate systems are quite insensitive to various mesh distortions,and have small computational amount.The element can simulate strain gradient effects of both nearly incompressible and compressible Cosserat continuum.The new elements can be applied to couple stress plastic theory;it also can be applied to generalized conforming element and incompatible element based on Cosserat theory.The new method provides a new thought to finite element method based on elastic Cosserat bodies.