基于辛几何法分析连铸板坯的黏弹性鼓肚变形

Viscoelastic Bulging Deformation Analysis of Continuous Casting Slab Based on the Symplectic Geometric Method

对板坯连铸机中板坯(坯壳)的鼓肚变形问题,考虑了坯壳在高温下呈现黏弹性性态以及弹性变形与黏性变形的耦合特性,提出了连铸板坯鼓肚变形的四边固支黏弹性矩形板模型。根据Laplace变换和弹性-黏弹性相应原理,导出了黏弹性板柸在拉氏域内的四阶偏微分方程,再将其导入拉氏域内的哈密顿正则方程。对全状态矢量利用分离变量法、本征函数展开法、共轭辛正交归一化关系和Laplace逆变换,得到了四边固支黏弹性矩形坯壳在时域内的挠度解析解。以某板坯连铸机为例,分析了弹性变形、蠕变变形和松弛时间对鼓肚变形的影响,以及坯壳中点的单位长度弯矩随时间的变化情况。

For the bulging deformation of the slab shell in slab continuous casting machine, the viscoelastic behaviors of the slab shell under high temperature and its coupling characteristics of elastic deformation and viscous deformation are considered, and a model of viscoelastic rectangular plate with four clamped edges for the bulging deformation of the slab shell is presented. Based on the Laplace’s transform and elastic-viscoelastic corresponding principle, the partial differential equation of four order of viscoelastic plate in the Laplace domain is derived, and then transferred into Hamilton canonical equation in the Laplace domain. For whole state variables, using the separation of variables, eigenfunction expansion in the symplectic geometry, conjugate symplectic orthogonal normalized relations and Laplace’s inverse transform, the analytical solution of the deflection for the viscoelastic slab with four edges clamped is obtained in time domain. Taking a slab continuous casting machine as an examples, the effect of elastic deformation, creep deformation and relaxation time of the material on the bulging deformation are analyzed, and the bending moment per unit length of the middle point for the solidified shell versus time are given.