The Refined Nonconforming Element Analysis Of Couple Stress/Strain Gradient Theory

There are extensive researches and applications of micro/nano-devices, along with the development of science and technology. At present, numerous experiments have demonstrated strong size effects on mechanical properties of materials at the micro/nano-scale. The micro/nano device design faces a series of new challenges posed by size effects. The classical continuum mechanics is not applicable for revealing the size effects of materials due to the lack of any material parameter corresponding to internal length scale in the constitutive models. Gradient theories can successfully explain the size effects by introducing the material length parameters with the length dimensions into the constitutive models, and have been widely used in the mechanical analysis of metal, granular and composite materials at the micro/nano-scale.This paper is focused on the couple stress theory and the strain gradient theory, which are two kinds of typical and widely used gradient theories. Compared with the conventional continuum mechanics, the couple stress/strain gradient theory is substantially more complicated, and heretofore only a few analytical solutions are available. Finite element method provides an important approach. Reliable finite element method is needed not only for the purpose of engineering applications, but also for the identification of the material length parameters where higher order accuracy is required. In the finite element analysis, the displacement interpolation function should satisfy the requirement of C1 continuity as first and second derivatives of the displacement are involved in the couple stress/strain gradient theory. C1 conforming elements contain the nodal parameters with high order derivatives, and are complicated to construct and implement. Further, there are few C1 conforming elements available. Currently, the most widely used couple stress/strain gradient elements are C0 elements, in which displacements and displacement gradients are interpolated independently and their kinematic constraints are enforced via the penalty or Lagrange multiplier method. However, it is difficult to identify the penalty function, and the Lagrange multiplier method may increase the computation cost. The methods of finite element construction and convergence test for the couple stress/strain gradient theory have not been fully developed.Compared with the conforming element methods, it is easier for the nonconforming element methods to establish high-performance elements as they relax the continuity condition more loosely and offer more flexible interpolation algorithms. The existing couple stress/strain gradient elements are constructed based on the consideration of either C0 or C1 continuity. In this paper, the refined nonconforming finite element methods are used to establish the plane and axisymmetric couple stress/strain gradient elements which satisfy C0 continuity (or weak C0 continuity) and C1 continuity simultaneously. Firstly, a 24-DOF (degrees of freedom) quadrilateral refined nonconforming element (CQ12+RDKQ) for the couple stress/strain gradient theory is developed. An extended Hu-Washizu variational principle which relaxes the continuity condition is proposed. Based on this variational principle, a 12-DOF quadrilateral nonconforming element CQ12 which satisfies weak C0 continuity and quadratic completeness is developed to calculate strains. The strain gradients are computed by the 12-DOF thin plate element RDKQ, which satisfies weak C1 continuity. By combining RDKQ and CQ12, and replacing the parameters of plate element by those of plane element, the 24-DOF element (CQ12+RDKQ) is established. Secondly, an 18-DOF axisymmetric triangular refined nonconforming element (BCIZ+ART9) for the couple stress/strain gradient theory is derived. Up to now, only a few axisymmetric couple stress/strain gradient elements have been developed. The axisymmetric C1 element does not exist. In this paper, a weak C1 continuity condition of axisymmetric nonconforming element method is proposed, and furthermore, the axisymmetric element (BCIZ+ART9) is developed. The displacement function of BCIZ, which satisfies C0 continuity and quadratic completeness, is used to calculate first derivatives of displacement. And the displacement function of ART9, which satisfies the proposed C1 weak continuity condition, is used to calculate second derivatives of displacement.In finite element analysis, the patch test has been used as a criterion for assessing the convergence of finite elements. The equilibrium differential equations of couple stress/strain gradient theory are inhomogeneous, and the conventional patch test functions are not appropriate for such kind of problems. In this paper, based on the C0-1 patch test and the enhanced patch test, the patch test function for axisymmetric couple stress element is established, and furthermore, it is proved that the constant shear stress patch test function does not exist for conventional axisymmetric elements. The proposed elements (CQ12+RDKQ) and (BCIZ+ART9) can both pass the patch test (ensure convergence) and possess high-performance.Finally, utilizing the proposed elements, the elastic process of the reinforcement pull-out and the deformation of a cantilever beam are simulated based on two kinds of strain gradient theories in which the signs of higher-order differential terms are opposite. The numerical results show the difference between the two theories in the aspect of describing size effects of materials.