一种求解三向应力状态主方向的方法

A method for solving the direction of principal stress in 3-D stress state

为了简单地确定三向应力状态的主方向,根据应力张量是一个实对称矩阵,利用实对称矩阵的特征值与特征向量之间的关系来研究主应力的方向。若应力张量的特征值(主应力)互不相等,则在数学上保证了特征向量(主方向)正交;若应力张量有相同的主应力,则会出现应力主方向非正交的情况。非正交的主方向对应圆柱体侧面或球体受压或受拉的情况,可采用施密特正交单位化处理,使得主应力方向在代数上的求解方式更为规范。应用具体算例说明了新方法的可行性和在代数上的严密性。

To simply determine the principal direction of the three-dimensional stress state,according to the fact that the stress tensor is a real symmetric matrix,the direction of the principal stress was studied by using the relationship between the eigenvalues and eigenvectors of the real symmetric matrix.If the eigenvalues(principal stresses)of the stress tensors are not equal to each other,then the eigenvectors(principal directions)are orthogonal in algebra.If the stress tensors have the same principal stresses,the principal directions of the stresses will be non-orthogonal.The non-orthogonal principal directions correspond to the compression or tension of the side of the cylinder or the sphere.Schmidt’s orthogonal unitization can be used to deal with the principal stress directions.The algebraic solution of the principal directions is more standardized.The feasibility and algebra tightness of the new method was illustrated by an example.