Index notation for tensors and vectors
Index notation is used extensively in literature when dealing with stresses, strains and constitutive equations. The reason is that it reduces drastically the number of terms in an equation and simplifies the expressions. We will use a right handed Cartesian coordinate system to describe the index notation (cf. Fig. 1). Moreover it is more convenient to name the axes \( x_{1} \), \( x_{2} \) and \( x_{3} \) instead of the more familiar notation \( x \), \( y \), \( z \).
Mohr's circle in 3 dimensions
Mohr's diagram is a useful graphical representation of the stress state at a point. In this graphical representation the state of stress at a point is represented by the Mohr circle diagram, in which the abscissa \( \sigma \) and \( \tau \) give the normal and shear stress acting on a particular cut plane with a fixed normal direction. In the general 3 dimensional case, for a given state of stress at a point, the Mohr circle diagram has three circles as shown in Fig. 1. Mohr's circle diagram is used frequently in conjunction with failure criteria like …
Octahedral stresses
Octahedral stresses we call the normal and shear stresses that are acting on some specific planes inside the stressed body, the octahedral planes. If we consider the principal directions as the coordinate axes (see also the article: Principal stresses and stress invariants), then the plane whose normal vector forms equal angles with the coordinate system is called octahedral plane. There are eight such planes forming a regular octahedron as it is illustrated in Fig. 1.
Deviatoric stress and invariants
The stress tensor can be expressed as the sum of two stress tensors, namely: the hydrostatic stress tensor and the deviatoric stress tensor. In this article we will define the hydrostatic and the deviatoric part of the stress tensor and we will calculate the invariants of the stress deviator tensor. The invariants of the deviatoric stress are used frequently in failure criteria.
Consider a stress tensor \( \sigma_{ij} \) acting on a body. The stressed body tends to change both its volume and its shape. The part of the stress tensor that tends to change the volume of the body …
Principal stresses and stress invariants
In this article we will discuss the derivation of the principal stresses and the stress invariants from the Cauchy stress tensor. The principal stresses and the stress invariants are important parameters that are used in failure criteria, plasticity, Mohr's circle etc.
For every point inside a body under static equilibrium there are three planes, called the principal planes, where the stress vector is normal to the plane and there is no shear component (see also: Calculation of normal and shear stress on a plane). These normal stress vectors are called principal stresses. From the mathematical point of view, the …