By C. Castellani
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Additional info for Disordered Systems and Localization
7 The Isotropic Tensors A tensor is isotropic if it has the same components with respect to a Cartesian coordinate system of any orientation. Isotropic tensors of various ranks are given below: Scalar: Every scalar is an isotropic tensor, because the scalar does not have a direction. Vector: There is no nontrivial isotropic vector. 122) where λ is a constant. 124) where α, β, and γ are constants. 129) and we see that the independent constants are now reduced to α and β. 7 Show that there are no nontrivial isotropic vectors.
These are intensity of forces (force per unit area) along the coordinate directions. They vary with the location in the body and their magnitudes and speciﬁc characteristics depend on the geometrical shape of the body and the loading condition (surface traction). To determine the distribution of stress in the body, we need to solve a boundary-value problem, by specifying the boundary conditions. However, this problem will not be discussed in this chapter. Instead, we concentrate on understanding the nature of stress at a given point, and consider the stress at a point.
153) where ds is the length of dxi . 155) Cartesian Tensors 39 Note that ni ui is negative at A and positive at B. 158) is valid for the inﬁnitesimal cylinder considered. 150). We remark that the theorem of Gauss is still valid for multiply connected region. In that case, we have to make branch cuts to prove the theorem. Some special cases of the theorem are listed below: 1. 7 Contour and surface. 151). 152) by ekij . 2. 165) In the two-dimensional case, the theorem changes the surface integral into the line integral and vice versa.
Disordered Systems and Localization by C. Castellani