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基于微态理论与应变梯度弹性理论框架,对含约束薄膜的单轴拉伸问题进行了研究。推导出在不同微观约束边界条件下薄膜单轴拉伸的解析解,较好的预测了薄膜内的边界层效应。通过分析两种理论之间的内在联系,发现可选取微态理论中的耦合因子作为罚参数,使得微态理论可以退化至应变梯度弹性理论。计算结果表明施加罚参数后的有限元解在边界层区域外与应变梯度弹性解析解吻合较好,即由于耦合因子的罚参数特性,使得基于微态理论开发的有限元程序可以应用于应变梯度弹性理论的模拟解答。
Based on the theory of microstrain and the theory of strain gradient elasticity, the uniaxial tension of the constrained film was studied. The analytical solution of uniaxial tension of the film under different microscopic confinement boundary conditions is deduced, and the boundary layer effect in the film is predicted well. By analyzing the intrinsic relations between the two theories, we find that the coupling factor in the microstates theory can be selected as the penalty parameter so that the microstrain theory can degenerate to the strain gradient elasticity theory. The calculated results show that the finite element solution with penalty parameter fits well with the analytical solution of strain gradient elasticity outside the boundary layer. That is to say, due to the penalty parameter of the coupling factor, the finite element program developed based on the theory of microstrain can be applied to strain gradient Simulation of elasticity theory.