Notes on Stellar Statistics II The Mathematical Expression by Luyten W.J.

By Luyten W.J.

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2 Local and global fracture parameters A correct transition from the microscopic description to the macroscopic one is a complicated and not uniquely defined problem. On the one hand, it is necessary to take into account the properties of elementary structural units (for example, the existence of the grain boundaries in steel). On the other hand, it is important to obtain only integral (averaged) features of the medium. Besides, according to the macroscopic approach, it is necessary to have some parameter connected with the microscopic scale.

For example, the Cosserat medium with rotation restrictions corresponds to second-order materials in the elasticity theory of micropolar materials. For the second branch, an expression in the space of stresses of the torsion tensor is nothing more but a description of couple stresses; therefore, respectively, they are linked with the elasticity theory of micropolar materials (the first branch) and the continuum nonlocality theory (the third branch of Fig. 2). The statistical description (method of correlation functions) of structure and properties of solids is standing somewhat apart [58].

According to the general rule of vector decomposition by basis set, we may write down: ξ 0 = ξ i (s)xi (x 1 , . . , x n ). 1. It is necessary to take into account that arguments x 1 , . . , x n themselves depend on the natural parameter. 20) with account that ξ 0 = const, we have: 0 = dξ i xi + ξ i dxi . 21) According to the usual formula for total differential dxi (x 1 , . . 22) where ∂ 2 x(x 1 , . . , x n ) . 23) ∂x i ∂x j Since xij is a vector, it may be resolved into vectors of the local reference field xi in an arbitrary point: xij = xij = k 1 n ij (x , .

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