S-R分解定理的张量表述及其率形式

The Tensor Form and Rate Form of S-R Decomposing Theorem

目的研究S-R分解定理导出的应变率及旋转率,进一步完善S-R理论.方法以实方阵的对称、正交和分裂定理为基础,推导大变形、大转动情况下连续介质变形梯度张量的S-R分解定理,结合张量理论研究参数sinθ的本质及其在非笛卡尔坐标系下的计算公式,利用连续介质力学分析S-R分解定理前提条件、几何意义及由S-R分解定理导出的应变率和旋转率.结果由S-R分解定理导出的应变率、旋转率与通常使用的应变率、旋转率线性部分相同.结论将参数sinθ以张量形式表达可使S-R分解定理应用于非笛卡尔坐标系,同时需严格满足其前提条件.S-R理论体现了工程线应变、工程剪应变对应变张量S和局部刚体转动张量R的共同依赖性.

In order to improve S-R decomposing theorem and investigate strain rate and rotation rate educed from S-R decomposing theorem,based on direct division theorem for real square matrix to symmetric matrix and orthogonal matrix, the derivation and tensor form of S-R decomposing theorem for continuous medium in the case of large deformation and rotation are illuminated. The essence of parameter sin0 is analyzed for ensuring isotropic property of S-R decomposing tensor equation and its computational formula in non-Des- cartes coordinate system is indicated. Combined with continuous medium mechanics, the precondition and geometric meaning of S-R decomposing theorem are presented. Meanwhile, the strain rate and rotation rate e- duced from S-R decomposing theorem are researched, linear parts of strain rate educed from S-R decompo- sing theorem and Green deformation are coincident. S-R decomposing theorem indicates that line-strain and shear-strain depend on both deformation and local rigid rotation.