Theoretically, the constant stress patch test is not rigorous. Also, either the patch test of non-zero constant shear for Mindlin plate problem or non-zero strain gradient curvature of the microstructures cannot be performed. To improve the theory of the patch test, in this paper, based on the variational principle with relaxed continuity requirement of nonconforming element for homogeneous differential equations, the author proposed the individual element condition for passing the patch test and the convergence condition of the element: besides passing the patch test, the element function should include the rigid body modes and constant strain modes and satisfy the weak continuity condition, and no extra zero energy modes occur. Moreover, the author further established a variational principle with relaxed continuity requirement of nonconforming element for inhomogene-ous differential equations, the enhanced patch test condition and the individual element condition. To assure the convergence of the element that should pass the enhanced patch test, the element function should include the rigid body modes and non-zero strain modes which satisfied the equilibrium equations, and no spurious zero energy modes occur and should satisfy new weak continuity condition. The theory of the enhanced patch test pro-posed in this paper can be applied to both homogeneous and inhomogeneous differential equations. Based on this theory, the patch test of the non-zero constant shear stress for Mindlin plate and the C0-1 patch test of the non-zero constant curvature for the couple stress/strain gradient theory were established.
A new higher-order shear deformation theory based on global-local superposition technique is developed. The theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. The global displacement components are of the Reddy theory and local components are of the internal first to third-order terms in each layer. A two-node beam element based on this theory is proposed. The solutions are compared with 3D-elasticity solutions. Numerical results show that present beam element has higher computational efficiency and higher accuracy.
基于整体-局部位移方法,建立了一种高阶剪切变形理论。整体位移部分采用的是R eddy理论的位移模式(1984),局部位移为L I X Y等(1997)建立的1,2-3理论的局部函数。这一理论使满足自由表面条件的R eddy理论进一步满足层间位移、应力连续,同时有效减少了1,2-3理论的未知数个数。基于此理论深入开展了有限元法研究,建立了满足C1连续条件的精化三节点三角形单元(每个节点参数为9个)。计算结果表明:建立的精化单元能准确计算整体位移和层间应力。
