中文摘要：

自由边问题一直是三维弹性力学中的难题，通常很难满足自由边上一个正应力和两个剪应力都等于0．基于三维弹性力学基本方程和状态空间方法，引入自由边界位移函数并考虑全部弹性常数，建立了正交异性具有自由边单层和叠层板的状态方程．对状态方程中的变量以级数形式展开，通过边界条件的满足精确求解任意厚度具有自由边叠层板的位移和应力，此解满足层问应力和位移的连续条件．算例计算表明，采用引入的位移函数形式，简化了计算过程并且采用较少的级数项可以获得收敛解．与有限元方法计算结果进行了对比，可以得到较高精度的数值结果．其解可以作为其它数值方法和半解析方法的参考解．

英文摘要：

The problem of free-edges in three-dimensional elasticity is always a difficult one. The conditions that both normal stress and shear stress on the free edges equal zero are satis- fied very difficultly. Based on the three-dimensional fundamental equations of elasticity and the state space method, the state equation for orthotropic plates was established through introduc- tion boundary displacement function and consideration of all elastic constants of the orthotropic materials. Series expansion was carried out on the variables of the state equation. An exact so- lution was presented for laminated plates with arbitrary thickness by satisfaction of boundary conditions, which could also satisfy the continuous conditions of stresses and displacements be- tween plies of the laminates. The results of two examples show that the calculation process is simplified and the convergent solution can be achieved with less terms of series when the dis- placement function of free boundary is adopted. The numerical results with high accuracy can be obtained through comparison with the results of finite element. The results can be used as reference to numerical methods and semi-analytical methods.