Main Equations in Media Mechanics

Abstract

The stresses p, σ, τ for tension and compression were introduced in Chap. 1. In order to study the stress state in a point the cube is usually cut about it (Fig. 2.1). Acting on its faces stresses are often written in a form of a matrix which is called the tensor (of stresses) as

$${\rm T}_{\rm \sigma } = \left( {\begin{array}{*{20}c} {\sigma _{{\rm xx}} } & {\tau _{{\rm xy}} } & {\tau _{{\rm yz}} } \\ {\tau _{{\rm yx}} } & {\sigma _{{\rm yy}} } & {\tau _{{\rm yz}} } \\ {\tau _{{\rm zy}} } & {\tau _{{\rm zy}} } & {\sigma _{{\rm zz}} } \\ \end{array}} \right)$$

(2.1)

where τ_ij = τ_ji and as a rule it is supposed σ_xx ≡ σ_x, σ_yy ≡ σ_y, σ_zz ≡ σ_z. There is an interesting interpretation of a tensor as the vector in 9-dimensional space /7/ the part of which is shown in Fig. 1.22.