Principal invariant

Partial derivatives of the principal invariants and moments of A are given by... 

In general, /xa is a complex and often nonlinear function of IId, the second principal invariant of D Hd is given by... 

Because the viscosity is a scalar, it can be a function only of the scalar invariants of the rate of deformation tensor. There are three combinations of the components of the rate of deformation tensor (Ajj), which are scalar invariants. They define any state of deformation rate independentiy of the coordinate system. They are referred to as the principal invariants of the rate of deformation tensor ... 

If a fluid can be considered incompressible, the first principal invariant of the rate of deformation tensor will be zero, Ij = 0. The third principal invariant vanishes in many simple flow situations, like axial flow in pipe, tangential flow between concentric cylinders, etc. In more general terms, the third invariant is zero in rectilinear... 

Apart from the principal stresses, a stress tensor has another set of invariants, called the principal invariants Ji, J2, and J3 (see appendix A. 7). They are defined as... 

The von Mises yield criterion uses the principal invariants of the deviatoric stress tensor, J(, J2, and J3. Using equation (3.26), we find... 

There is a special set of invariants, the principal invariants Jk, that can be formed from the eigenvalues ... 

These principal invariants are important in the context of yielding of materials and are used in section 3.3.1. 

Every scalar function lyofa symmetric tensor G which satisfies (27) is called an Isotropic Tensor Function of G. An isotropic scalar-valued function of G is also called a scalar invariant of G. It may easily be checked that the principal invariants of G, defined by... 

Where y/i,y/2 are scalar functions of time and of the first and second principal invariants, I1J2 of C- Then (129) reduces to... 

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