New notions of uniformity and homogeneity of Cosserat media

In this paper, we study internal properties of a Cosserat media. In fact, by using groupoids and smooth distributions, we obtain a three canonical equations. The $\mathit{\text{non-holonomic}}\mathit{\text{material}}\mathit{\text{equation}}\mathit{\text{for}}\mathit{\text{Cosserat}}\mathit{\text{media}}$ characterizes the uniformity of the material. The $\mathit{\text{holonomic}}\mathit{\text{material}}\mathit{\text{equation}}\mathit{\text{for}}\mathit{\text{Cosserat}}\mathit{\text{media}}$ permits us to study when a Cosserat material is a second-grade material. It is remarkable that these two equations also provide us a unique and maximal division of the Cosserat medium into uniform and second-grade parts, respectively. Finally, we present a proper definition of homogeneity of the Cosserat medium, without assuming uniformity. Thus, the $\mathit{\text{homogeneity}}\mathit{\text{equation}}\mathit{\text{for}}\mathit{\text{Cosserat}}\mathit{\text{media}}$ characterizes this notion of homogeneity.