轴对称弹性应变梯度理论公式推导及有限元实现

Formula derivation in axisymmetric strain gradient theory and finite element implementation

用张量运算推导了弹性应变梯度轴对称问题的基本公式。建立了应变梯度轴对称不协调元的弱连续条件,进一步建立了满足弱连续条件的应变梯度轴对称18-DOF三角形单元(BCIZ+ART9),其中BCIZ满足线性应变C0连续,用于计算应变ε;ART9满足常曲率C1弱连续,用于计算应变梯度η。数值结果表明该单元通过C0-1分片检验并能体现材料的尺度效应。

This paper deals with the basic functions of axisymmetric strain gradient theory. A weak continuity condition, which assured the axisymmetric strain gradient element convergent is proposed. And an 18-DOF axisymmetrie refined nonconforming triangular element （BCIZ＋ARTg）is derived. The displacement function of the thin plate triangular element BCIZ, which can pass the C^0 linear stress patch test, is used to calculate strain ε. And the element function of the refined element ARTg, which is derived from BCIZ and satisfies the interelement C^1 weak continuity conditions, is used to calculate the strain gradient η. Numerical results show that the element can reflect the scale effects of strain gradient and pass C^0-1 patch test.