缺陷连续统理论及其在本构方程研究中的应用 Ⅱ.缺陷的规范场理论

CONTINUUM THEORY OF DEFECTS AND ITS APPLICATION TO THE STUDY OF CONSTITUTIVE EQUATIONS Ⅱ. Gauge Field Theories of Defects

详细评述了缺陷连续统的规范场理论,该理论是近代材料科学和固体力学中新发展起来颇有意义的一个分支。首先强调了Noether定理及其逆定理在构造缺陷规范场理论中的重要性。同时基于Yang-Mills普适规范场构造,包括对SO(3)T(3)群的最小替换和最小耦合原理,系统地介绍了Golebiewska-Lasota,Edelen,Kadic和Edelen等人的原始性工作及他们的贡献。本文表明,Kadic和Edelen的理论是基于一组缺陷动力学的线性连续性方程发展起来的,不能和关于缺陷场的现有几何理论完全协调起来。考虑到这一点,本文提供了另一种方法来建立非线性弹性规范场的相应理论,这里考虑了Poincaré规范群SO(3)T(3).采用类似于研究引力场理论的Kibble方法,导出了缺陷连续统的拉氏密度。非完整坐标变换和非欧联络系数在数学上完全等价于子Poincaré群SO(3)T(3)的规范场。因此,本文的规范场理论和4维物质流形的缺陷场的非线性几何理论是完全一致的,并证明在弱缺陷条件下,可以简化到Kadic和Edelen的结果。

The gauge field theory of defects as a newly-developed and emerging branch in modern solid mechanics and material science is reviewed in detail. Noether's theorem and its inverse theorem, which play a significant role in the construction of the defect gauge theory is first introduced. The original works and contributions due to Golebicwska-Lasota, Edelen, Kadic and Edelen, etc, are systematically brought in on the basis of Yang-Mills universal gauge theory construction, including the minimal replacement principle and minimal coupling principle for the group SO(3)(?)T(3). Because Edelen,Kadic and Edelen's theories arc developed by making use of a special set of linear continuity equations of defect dynamics, the theories seem not to be completely in harmony with the existing geometrical theory of gauge field. Considering this fact, an alternative way has been tried to establish a corresponding theory of a nonlinear elastic gauge field, where the sub-Poincare gauge group SO(3)(?)T(3) are taken into account to replace the mere gauge group SO(3)(?)T(3) as adapted by Kadic and Edelen. Using a similar way as given by Kibblc in his study of gravitation field theory, the Lagrangian density for the defect continuum is obtained. Because the anholonomic coordinate transformation and non-Euclidean connection coefficients of the moving frame in the natural state are shown to be equivalent in their geometrical structure to the representation of the sub-Poincare group, the present gauge theory of defects is completely consistent with the nonlinear geometric theory of defect field within the framework of 4-dimensional material manifold M4, and it can be reduced to the theory of Edelen, Kadic and Edelen in the case of small deformation and weak defect approximation.