厚壁筒的一种简化弹性应变梯度模型

Simplified elastic strain gradient model of thick hollow cylinder

在Toupin-Mindlin应变梯度理论框架内,建立了一种关于厚壁筒的应变梯度模型,并利用打靶法和拟牛顿迭代法实现了两点复杂边界条件和四阶常微分平衡方程的求解,得到了厚壁筒应力场和位移场精确数值解。为了降低该模型的复杂性和提高该模型的适用性,通过对高阶应力做出简化假设,得到了简化的平衡方程和相应的厚壁筒应力、应变的解析解。通过对厚壁筒经典弹性理论解析解、梯度理论简化解和梯度理论精确数值解的比较,检验了简化模型解析解的可行性和精确性,并得到了梯度理论和经典理论在厚壁筒应用上的差异性。

The simplified equilibrium equation and analytical solutions to strains and stresses in thick hollow cylin-der are obtained in the framework of Toupin-Mindlin′s strain gradient theory by adopting simplifying assumptions with respet to the higher-order stresses. To verify the feasibility of the analytical solutions of the simplified model, Quasi-Newton method, together with a shooting method,was used to obtain accurate numerical solutions of the origi-nal fourth-order ordinary differential equilibrium equation with complex boundary conditions. The analytical solu-tions of conventional elastic theory, analytical solutions of the simplified gradient model and the accurate numerical solutions of the original gradient model were compared to test the feasibility and accuracy of the simplified gradient model and to acquire the differences between the original gradient theory and the conventional elastic theory in the thick hollow cylinder.