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Additional resources for 1/N expansion for scalar fields
It is always possible to calculate components of the stress tensor for any direction and to find principal directions and principal normal stresses; the latter A. Ya. Malkin 33 are extremal and shear stresses are absent at those directions. There are three special combinations of any arbitrary stress tensor which do not depend on choice of axes orientation in space. These combinations are called invariants. The stress tensor can be divided into two parts, one of which (so-named spherical) is responsible for volume and the other (called deviator) for shape (or form) changes of a body (at a point).
This is completely true for the rate of deformation. 13. Moreover, in some theoretical studies, there is a necessity to introduce kinematic tensors of a higher order hich are determined as time derivatives of the Cauchy-Green or Finger ten sors. They are used if it is assumed that rheological behavior of a material depends, not on the rate of deformation only, but on higher derivatives of deformation, as well. In Chapter 7, devoted to properties of viscoelastic materials, their behavior is modeled by equations containing a sum of n-th order time derivatives of deformation (so-called rheological equations of a differential type).
A circular hole weakening a flat sheet. Since practically all parts of engineering and everyday applications have holes of different sizes and shapes, it is very important to know how these holes influence performance characteristics of a part. In technical language, this influence is associated with stress fields appearing in the neighborhood of a hole. A round (circular) hole is the simplest example of possible shapes of holes, though holes can have different shapes. If a sheet having a circular hole is stretched by a uniform load, p, stress is created around the hole.